on primitive overgroups of quasiprimitive permutation groups

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Journal of Algebra 263 (2003) 294–344

www.elsevier.com/locate/jalgebr

On primitive overgroups of quasiprimitivepermutation groups

Robert W. Baddeleya and Cheryl E. Praegerb,∗,1

a Black Dog Software, 32 Arbury Road, Cambridge CB4 2JE, United Kingdomb Department of Mathematics and Statistics, The University of Western Australia,

35 Stirling Highway, Crawley, Western Australia 6009, Australia

Received 10 October 2002

Communicated by Jan Saxl

Abstract

A permutation group is said to bequasiprimitive if all its non-trivial normal subgroups artransitive. We investigate pairs(G,H) of permutation groups of degreen such thatG H Snwith G quasiprimitive andH primitive. An explicit classification of such pairs is obtained excepthe cases where the primitive groupH is either almost simple or the blow-up of an almost simgroup. The theory in these remaining cases is investigated in separate papers. The results dthe finite simple group classification. 2003 Elsevier Science (USA). All rights reserved.

Keywords:Permutation groups; Primitive groups; Quasiprimitive groups

1. Introduction

A permutation group is said to bequasiprimitiveif all its non-trivial normal subgroupare transitive. Thus any primitive permutation group is quasiprimitive, and quasiprimiis a natural weakening of the condition of primitivity. We were told in January 199Wolfgang Knapp that Helmut Wielandt had suggested that the term ‘quasiprimitiv

* Corresponding author.E-mail addresses:[email protected] (R.W. Baddeley), [email protected]

(C.E. Praeger).1 For the second author this paper forms part of a research project supported by an Australian Research

large grant.

0021-8693/03/$ – see front matter 2003 Elsevier Science (USA). All rights reserved.doi:10.1016/S0021-8693(03)00113-3

R.W. Baddeley, C.E. Praeger / Journal of Algebra 263 (2003) 294–344 295

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moreof [31]

itiveatrix

, HS,. Letcan

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–4.5,

used for this concept, and to our knowledge it first appeared in papers of Knapp1970s, for example, in [18,19].

The immediate aim of this paper is to describe, in so far as it is possible, all embedof a quasiprimitive permutation group into a primitive one. Thus this paper can be sea direct generalisation of [31]. However, this paper should also be seen in the wider cof an investigation into the suitability of quasiprimitivity as a reduction tool in applicatof group theory. This investigation is two-pronged; it involves, on the one hand, an atto understand quasiprimitive permutation groups in a purely group-theoretic setting, athe other, actual applications of quasiprimitivity to classification problems, in this insto problems concerning 2-arc transitive graphs [2,8,9,13,14,21,22,32]and linear spaSeveral of the proofs rely on the classification of the finite simple groups.

In order to describe our results we make use of the case distinction of primpermutation groups as given in [31]: this distinguishes eight types of primitive permugroups, namely types HA, HS, HC, AS, TW, SD, CD, and PA. These types are definSection 3, along with an analogous case distinction for the quasiprimitive permugroups. In the quasiprimitive case we again have eight types, and we label thesHS, HC, AS, TW, SD, CD, or PA. (We note that the quasiprimitive permutatigroups of type XX that are primitive are precisely the primitive permutation grouptype XX; furthermore all quasiprimitive permutation groups of types HA, HS, HCprimitive. Thus capital–capital denotes primitivity, whilst capital–small capital denquasiprimitivity with possible imprimitivity.)

Definition 1.1. The pair(G,H) is said to be aquasiprimitive inclusionif G H Snwith bothG andH quasiprimitive. It is called aquasiprimitive-primitive inclusionif, inaddition,H is primitive; and is called aprimitive inclusionif, in addition,G is primitive(and henceH is primitive also). It is an(X,Y )-inclusion, ifG is quasiprimitive of typeXandH is quasiprimitive of typeY . It is aproper inclusionif G is a proper subgroup ofH .

In this paper we describe quasiprimitive-primitive inclusions. Some features ofgeneral quasiprimitive inclusions are discussed in a sequel [33]. Given the resultswe are only interested in quasiprimitive-primitive inclusions(G,H) with G imprimitive;thusG is not of type HA, HS, or HC. To describe the possible quasiprimitive-priminclusions in the remaining cases we have constructed a ‘Results Matrix’. This is a mwith rows indexed by the types of quasiprimitive permutation groups other than HAand HC, and with columns indexed by the types of primitive permutation groupsR(X,Y ) be the entry in the(X,Y ) position. There are six categories of values thatbe taken byR(X,Y ), and each has a significance as follows:

− There are no(X,Y )-inclusions.P There exist(X,Y )-inclusions and they are all primitive.

1, . . . ,5 HereR(X,Y ) is a positive integer at most 5; the implication is that all pro(X,Y )-inclusions are as described by one of five explicitly given typesquasiprimitive-primitive inclusions; these are described in Sections 4.1respectively.

296 R.W. Baddeley, C.E. Praeger / Journal of Algebra 263 (2003) 294–344

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evantesend we

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tiononsent

HA HS HC AS TW SD CD PA

AS P 3 − 7 − 3 − 5

TW − − 3 − 2 3 3 6

SD − − − − − 1 − −CD − − − − − − B B

PA P − − − − 4 B 6

Fig. 1. The Results Matrix.

B This means that all proper(X,Y )-inclusions are ‘blow-ups’ of other quasprimitive-primitive inclusions; such inclusions are discussed in Section 4.6.

6 Here the Results Matrix makes no claims about the relevant inclusions;(−,PA)-inclusions are the subject of a separate work [4].

7 Here the Results Matrix makes no claims whatsoever about the relinclusions; (AS,AS)-inclusions are discussed in Section 7.1; work on thexamples is being undertaken by Liebeck, Saxl and the second author, abelieve that this will lead to a complete classification of them.

Theorem 1.2. Let (G,H) be a quasiprimitive-primitive inclusion of type(X,Y ) on a setof sizen such thatH = An or Sn andX = HA, HS or HC. ThenR(X,Y ) is as given in theResults Matrix.

Remark 1.3.

(1) In the cases where we haveR(X,Y ) = P, the only proper(X,Y )-inclusions are thoslisted in Table 1 of [31].

(2) The concept of a ‘blow-up inclusion’ requires the concept of a ‘blow-up opermutation group’ due to Kovács [20]; the latter will also be discussed in Sectio

(3) In the case of(−,PA)-inclusions the theory is surprisingly rich and a great deainformation is deduced in [4]. A brief discussion of the relevant results of [4] is gin Section 4.6.

(4) The fact that there are no(X,AS) inclusions(G,H) with H = An or Sn andX ∈TW, SD, CD, PA follows from Theorem 1.4 below.

(5) For X = SD or CD and Y = CD or PA, the proof that the(X,Y )-entry of theResults Matrix is valid depends on [4]. (See Remark 4.11 for further informaabout(CD,CD) and(CD,PA) inclusions.) Also the proof of Theorem 1.4 depends[28]. Apart from this the proof of Theorem 1.2 is completely contained in the prepaper.

Theorem 1.4. Let H be an almost simple group, that is,S H Aut(S) for a non-abelian simple groupS, and suppose thatH = AB whereA is a proper subgroup ofHnot containingS, andB ∼= T k for a non-abelian simple groupT and integerk 2. ThenS = An andA∩ S = An−1, wheren = |H : A| = |S : A∩ S| 10.


someroupstion 4thuss theion 7hreeions to

t

The layout of the paper is as follows. Section 2 contains notation, terminology, andpreliminary results. Section 3 defines the eight types of quasiprimitive permutation gand constructs some explicit examples of quasiprimitive permutation groups. Secdescribes ‘inclusions of types 1–5’ and what is meant by a ‘blow-up inclusion’,clarifying the significance of the entries of the Results Matrix. Section 6 containproof of Theorem 1.2, and Section 5 contains the proof of Theorem 1.4. Sectconsiders(AS,AS)-inclusions and gives some closing remarks. In particular tproblems concerning factorisations of almost simple groups are posed, the solutwhich would facilitate a description of such inclusions.

2. Notation and terminology

2.1. General notation

All groups and sets are finite. The identity element of a groupG will be written idG, ormore simply as id if no confusion arises. IfH is a subgroup ofG, then thecoreof H in G,written CoreGH , is the largest normal subgroup ofG that is contained inH ; thus

CoreGH =⋂g∈G

Hg.

The subgroupH of G is core-freein G if its core is trivial. ThesocleSocG of a groupG is the direct product of all minimal normal subgroups ofG. (Note that any two distincminimal normal subgroups of a group necessarily centralise each other.) IfA andB aresubgroups ofG, then we useAB to denote the set given by

AB = ab: a ∈A, b ∈B.

We will often use the well-known result that

|AB| = |A||B||A∩B| .

If A, B are proper subgroups ofG satisfyingG = AB, thenA, B are said tofactoriseG,and the expressionG = AB is said to be afactorisationof G.

The group of all permutations of a setΩ will be denoted Sym(Ω); for Ω = 1, . . . , n,we use insteadSn. A permutation group on a setΩ is any subgroup of Sym(Ω). Let Gbe a permutation group onΩ . If x ∈ Ω , thenGx is thepoint-stabilizerin G of x and isdefined by

Gx = g ∈G: xg = x.

A permutation group isregular if it is transitive and all its point-stabilizers are trivial.


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The permutation groupsG on Ω andH on∆ arepermutationally isomorphicif thereexist a bijectionθ :Ω →∆ and an isomorphismχ :G →H such that

θ(xg) = θ(x)χ(g) for all x ∈ Ω, g ∈G.

If such conditions hold, the pair(θ,χ) is said to be apermutational isomorphism.Similarly, the pair(θ,χ) is an embeddingof the permutation groupG on Ω in thepermutation groupH on∆, if χ :G→ H is a monomorphism and(θ, χ) is a permutationaisomorphism, whereχ :G → χ(G) is obtained fromχ by simply restricting the range ofχ .

Note that if (θ,χ) is a permutational isomorphism fromG on Ω to H on ∆, thenχ

is an isomorphismG → H such that for allω ∈ Ω there existsδ ∈ ∆ with χ(Gω) = Hδ.Conversely, ifG is such thatχ(Gω) = Hδ for some isomorphismχ :G → H and someω ∈ Ω,δ ∈ ∆, then (θ,χ) is a permutational isomorphism betweenG and H , whereθ :Ω → ∆ is the bijection defined by

θ :ωg → δχ(g) for all g ∈ G.

On the other hand, ifθ is a bijection Ω → ∆ then θ induces an isomorphismιθ : Sym(Ω) → Sym(∆) given by

ιθ :π → θ−1 π θ for all π ∈ Sym(Ω),

where denotes composition of maps; we note that(θ,χ) is a permutational isomorphismfrom G Sym(Ω) to H Sym(∆) if and only if

ιθ (G) = H and χ = ιθ |G.

In particular, two subgroupsG, H of Sym(Ω) are permutationally isomorphic if and onif they are Sym(Ω)-conjugate.

Recall from Definition 1.1 that(G,H) is a quasiprimitive-primitive inclusion iG H Sn with G quasiprimitive andH primitive. We extend this definition tall permutation groups by saying that(G,H) is a quasiprimitive-primitive inclusion iG H Sym(Ω) for some setΩ with G quasiprimitive andH primitive on Ω . Twoquasiprimitive-primitive inclusions(G,H) and (G1,H1) are isomorphicif there existsa permutational isomorphism betweenH andH1 that restricts to give a permutationisomorphism betweenG andG1.

For a prime powerq , and integerr 2, a primitive prime divisorof qr − 1 is aprime which dividesqr − 1 but does not divideqi − 1 for any i < r. It was proved byZsigmondy [35] in 1892 thatqr − 1 has a primitive prime divisor unless either the p(q, r) is (2,6), or r = 2 andq = 2e − 1 is a Mersenne prime. We denote byqr a primitiveprime divisor ofqr − 1. Sinceqr ≡ 1 (modqr) andqi ≡ 1 (modqr) for i < r it followsthatq has orderr moduloqr , and in particularqr = ur + 1 r + 1 for some integeru.


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2.2. Subgroups of direct products

We will often be considering subgroups of direct products and wish to fix sterminology describing these.

Let M be the direct product of its non-trivial subgroupsT1, . . . , Tk, so that we have

M = T1 × · · · × Tk.

(The following definitions depend on this choice of direct decomposition ofM; althoughthe definitions do not assume so, it will always be the case in this paper thatM is non-abelian and characteristically simple, and that the direct factorsTi are the minimal normasubgroups ofM, and so are isomorphic non-abelian simple groups.) Letσi :M → Ti be thenatural projection map. LetK be a subgroup ofM. We say thatK is asubdirect subgroupof M if

σi(K) = Ti for all i = 1, . . . , k;

equivalently,K is asubdirect productof T1, . . . , Tk . We say thatK is adiagonal subgroupof M if

K ∼= σi(K) for all i = 1, . . . , k.

Moreover,K is a full diagonal subgroupof M, if it is both a subdirect and a diagonsubgroup ofM. Observe that full diagonal subgroups exist only if the factorsT1, . . . , Tk

are pair-wise isomorphic. We say thatK is a strip of M, if it is a diagonal subgroup othe direct product of some non-empty subset ofT := T1, . . . , Tk; equivalently, ifK isnon-trivial and, for eachi = 1, . . . , k, either

σi(K) ∼= K or σi(K) = id.

Moreover,K is a full strip of M, if it is a full diagonal subgroup of the direct productsome non-empty subset ofT = T1, . . . , Tk. For i ∈ 1, . . . , k and a stripK, we say thatK coversTi if σi(K) is non-trivial; thecovering setT (K) of K is defined as

T (K) = Ti ∈ T : K coversTi.

If |T (K)| > 1, then we say that the stripK is non-trivial. Two strips,K andL, aredisjointif their covering sets are disjoint. Note that disjoint strips commute.

The following easy result gives a link between subdirect subgroups and full strips

Lemma 2.1. LetM = T1×· · ·×Tk be the direct product of isomorphic non-abelian simgroupsT1, . . . , Tk . Suppose thatK is a subdirect product ofM. ThenK is the directproduct of pair-wise disjoint full strips ofM.


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Proof. This result follows from the lemma given in the appendix on maximal subgrcontained in [34].

The next lemma is a consequence of the classification of the (finite) simple group

Lemma 2.2. Suppose that

M = T1 × · · · × Tk

is the direct product of isomorphic non-abelian simple groupsT1, . . . , Tk . Suppose thaA1, . . . ,Aa are non-trivial pairwise disjoint strips ofM; setA = A1 × · · · × Aa . SupposethatB1, . . . ,Bb are also non-trivial pairwise disjoint strips ofM; setB = B1 × · · · × Bb.ThenM = AB.

Proof. Let T = T1 andn = |T |. It is clear that|A| na with equality if and only if eachof the stripsA1, . . . ,Aa are full strips; likewise|B| nb with equality if and only if eachof the stripsB1, . . . ,Bb are full strips. We assume thatM = AB, whence

nk = |M| |A||B| na+b.

It follows that a + b k. Given that the stripsA1, . . . ,Aa are disjoint and non-trivialthe covering setsT (A1), . . . ,T (Aa) are disjoint subsets ofT = T1, . . . , Tk, with eachcontaining at least two elements; thus

k ∣∣T (A1)

∣∣ + · · · + ∣∣T (Aa)∣∣ 2a.

Similarly we also havek 2b, whencek a + b. Equality follows betweenk anda + b,and in fact, we see that we have equality in all of the above inequalities. Hencek is even,a = b = k/2, |M| = |A||B|, and each stripAi andBj is a full strip and covers precisetwo of theTi . Note also that since|M| = |A||B|, we haveM = AB if and only if A∩B istrivial. By relabelling theT1, . . . , Tk if necessary we may assume that

T (Ai) = T2i−1, T2i for all i = 1, . . . , k/2,

and that for some), 1 ) k/2, we have

T (B1) = T2, T3, T (B2) = T4, T5, . . . , T (B)) = T2), T1,

if ) 2, orT (B1) = T1, T2 if ) = 1. LetM = T1 × · · · × T2), and write

M = (t1, . . . , t2)): ti ∈ T

,

so that forj = 1, . . . , ) we have


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yve

Aj =(t1, . . . , t2)) ∈ M: ti =

idT if i = 2j − 1 or 2j

tαj

2j−1 if i = 2j

and

Bj =(t1, . . . , t2)) ∈ M: ti =

idT if i = 2j or 2j + 1

tβj

2j if i = 2j + 1

,

where theαj andβj are automorphisms ofT . (Arithmetic in the above is modulo 2) sothat 2)+ 1= 1.) By straightforward computation, we find that

M ∩A∩B = (t, tα1, tα1β1, . . . , tα1β1···α)

): t ∈CT (α1β1 · · ·α)β))

.

It is a well-known consequence of the classification of the (finite) simple groups thatautomorphism of a non-abelian simple group centralises some non-trivial element (c4.1]), and soM ∩ A ∩ B, and so alsoA ∩ B, is always non-trivial. This contradictiocompletes the proof. 2.3. Automorphism groups and holomorphs

Let M be a group. We use AutM to denote the group of all automorphisms ofM,InnM the group of inner automorphisms, and OutM the quotient group AutM/ InnM.TheholomorphHolM of M is the semi-direct product

HolM = M AutM

formed with respect to the natural conjugation action of AutM onM. We callM thebasegroupof the holomorph.

There is a natural permutation action of HolM on its base group: we let the base groM HolM act by right multiplication, and AutM HolM by conjugation. Thus, if wewrite m ∗ x for the image ofm ∈M underx ∈ HolM, then we have

m ∗ yα = (my)α for all m,y ∈M, α ∈ AutM.

We call this thebase group actionof the holomorph. It is easy to verify that this actionfaithful, and so we can view HolM as a permutation group onM. Note that the base grouM is a regular normal subgroup of HolM onM.

Conversely, given a permutation groupG onΩ with a regular normal subgroupM, thenG is permutationally isomorphic to some subgroup of HolM in its base group action onM.More precisely, if we fixω ∈ Ω , then the regularity ofM means that the mapφ :Ω → M

given by

φ :ωm →m

is a well-defined bijection, and ifιφ : Sym(Ω) → Sym(M) is the isomorphism induced bφ, then by viewing HolM as a subgroup of Sym(M) via its base group action we hathat ιφ(G) is a subgroup of HolM with ιφ(M) equal to the base group of HolM and withιφ(Gω) = ιφ(G)∩ AutM. We leave the verification of this to the reader.


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2.4. Wreath products and twisted wreath products

We describe these two constructions for the purpose of explaining our notatioconventions concerning them.

Let G be any group and letH be a subgroup ofSn. We useGn to denote the direcproduct ofn copies ofG, i.e.,

Gn = (g1, . . . , gn): gi ∈ G

,

and defineG H , thewreath product ofG byH , to be the semi-direct productGnH inwhich the conjugation action ofH onGn is given by

(g1, . . . , gn)h−1 = (g1h, . . . , gnh) for all gi ∈ G, h ∈H.

If G0 andH0 are subgroups ofG andH respectively, then we will identifyG0 H0 withthe obvious subgroup ofG H .

We now turn to twisted wreath products, the concept of which was originallyto B.H. Neumann [29]. Here we follow the treatment of [2]. The ingredients forconstruction are:

a groupT , a groupP, a subgroupQ of P , and

a homomorphismφ :Q→ AutT .

DefineB to be the set of mapsf :P → T with multiplication defined point-wise. Therean action ofP onB given by

f p(x) = f (px) for all x ∈ P, f ∈ B;

let X be the semi-direct productBP with respect to this action. DefineBφ by

Bφ = f :P → T : f (pq) = f (p)φ(q) for all p ∈ P, q ∈Q

,

and observe thatBφ is a subgroup ofB normalised byP . The subgroupXφ = BφP of Xis called thetwisted wreath productof T by P with respect toφ, and we write

Xφ = T twrφ P.

We refer toP as thetop groupof Xφ , to φ as thetwisting homomorphism, to Bφ as thebase groupof Xφ , and toQ, the domain ofφ, as thetwisting subgroup. Note thatX isitself the twisted wreath product with respect to the trivial mapidP → AutT and thatBφ

∼= T k wherek = |P : Q|.Observe that there is a natural action ofT twrφ P on its base groupBφ in whichBφ acts

by right multiplication, andP acts by conjugation; we call this thebase group actionofthe twisted wreath product. (Compare with the base group action of a holomorph.)


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3. Quasiprimitive and primitive permutation groups

We start by recalling that ablock of imprimitivityfor a permutation groupG on a setΩis a non-empty subsetΓ of Ω such that for allg ∈G we have

either Γ g ∩ Γ = ∅ or Γ g = Γ.

A permutation groupG onΩ is primitive if and only if it is transitive and its only blockof imprimitivity areΩ and the singleton setsω for ω ∈ Ω ; equivalently, if and only if itis transitive and any point-stabilizer is a maximal subgroup ofG. As defined in Section 1a permutation group isquasiprimitiveif and only if all its non-trivial normal subgroups atransitive. It is elementary and well-known that any primitive group is quasiprimitive.

We define eight types of quasiprimitive permutation group analogous to the eightof primitive permutation group identified in [31]. The first three types are necessprimitive, and are permutationally isomorphic to primitive subgroups of the holomoof certain groupsK that are considered as permutation groups onK via the base grouaction, and that contain the socle of the holomorph. These are:

HA: such subgroups of theHolomorphof an Abelian group; these have a uniquminimal normal subgroupM (namely the base group of the holomorph), andM

is both regular and abelian.HS: such subgroups of theHolomorphof a non-abelianSimplegroup; these hav

precisely two minimal normal subgroupsM andN (namely the base group othe holomorph and the centralizer of the base group),M ∼= N , and bothM andNare regular, non-abelian and simple.

HC: such subgroups of theHolomorphof a Compositenon-abelian group; these haprecisely two minimal normal subgroupsM andN (namely the base group of thholomorph and the centralizer of the base group),M ∼= N , and bothM andN areregular and non-abelian, but are not simple. (These quasiprimitive permugroups are the blow-ups of those of type HS: see Remark 4.8.)

The five remaining types correspond to quasiprimitive permutation groups that mprimitive or imprimitive. We have

AS: anAlmost Simplegroup; such groups have a unique minimal normal subgroupM,andM is non-abelian and simple. The normal subgroup may be either regunot, and where appropriate we use ASreg and AS¬reg to distinguish the two.

TW: a Twisted Wreathproduct; such a group has a unique minimal normal subgM, andM is non-abelian and regular, but not simple.

Quasiprimitive permutation groups of the three remaining types have a unique mnormal subgroupM, and M is non-abelian, non-regular and non-simple; thusM isisomorphic to the direct productT k of k > 1 copies of some non-abelian simple groT . The types are distinguished by the nature of a point-stabilizer inM which is necessarilynon-trivial. (In the following we identifyM with T k .)


er

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HS,XXight

tionmitiveashionoof of

, HC,

p.

SD: a group ofSimple Diagonaltype; for such a group a point-stabilizer inM is a fulldiagonal subgroup ofM.

CD: a group ofCompound Diagonaltype; for such a group a point-stabilizer inM is adirect product of at least two disjoint non-trivial full strips ofM. (Groups of thistype are the blow-ups of groups of type SD: see Remark 4.8.)

PA: for such groups the identification ofM with T k can be chosen so that thpoint-stabilizer inM is a subdirect subgroup ofRk < T k = M for some propesubgroupR of T .

Theorem 3.1. The classes of quasiprimitive permutation groups as defined above arjoint and exhaustive; in other words, the type of any quasiprimitive permutation groudefined and is unique.

Proof. This follows from the results of [32]; the classes identified in [32] corresponthe present case distinction as follows:

I ←→ HAII ←→ AS

III(a)(i) ←→ SD

III(a)(ii) ←→ HSIII(b)(i) ←→ PA

III(b)(ii) ←→ HC and CD

III(c) ←→ TW We now identify eight types of primitive permutation groups, and label these HA,

HC, AS, TW, SD, CD, PA, by saying that the primitive permutation group is of typeif and only if it is of type XX as a quasiprimitive permutation group. Note that these etypes of primitive permutation group are precisely as defined in [31].

For more information on the structure of quasiprimitive and primitive permutagroups we refer the reader to [32] and [25] respectively. Note that the classes of pripermutation groups identified in [25] correspond to the present case distinction in a fexactly analogous to that for quasiprimitive permutation groups described in the prTheorem 3.1.

We now construct some examples related to quasiprimitive groups of types HSSD, and CD, that will be useful to us later.

Construction 3.2. Let m > 1, ) > 0 be integers and setn = m). If ) = 1 setH = Sm;otherwise letH be the maximal subgroup ofSn that has

1, )+ 1, . . . , )(m− 1)+ 1

, 2, )+ 2, . . . , . . . , ), . . . ,m)

as a system of imprimitivity. ThusH ∼= Sm S). Let T be a non-abelian simple grouDefine a subgroupW(T,m,)) of (AutT ) H as follows:


an

e

t any

eset

t

W(T,m,)) = (t1, . . . , tn)π : ti ∈ AutT , π ∈H, and

t−1i tj ∈ InnT if i ≡ j (mod))

.

Note thatW(T,m,)) contains(InnT )n as a unique minimal normal subgroup, and isextension of this by(OutT ×Sm) S). We identifyT with InnT so that Soc(W(T ,m,))) =T n. Also define the subgroup∆(T ,m,)) of W(T,m,)) by

∆(T ,m,))= (t1, . . . , tn)π ∈W(T,m,)): ti = tj if i ≡ j (mod))

.

The subgroup∆(T ,m,)) is a core-free subgroup ofW(T,m,)) as it does not contain thsocleT n; we letΩ(T,m,)) be the set of right cosets of∆(T ,m,)) in W(T,m,)) and viewW(T,m,)) as a permutation group onΩ(T,m,)) via its right multiplication action. IfT ,m and) are clear from the context, then we denoteW(T,m,)) simply byW , Ω(T,m,))

by Ω , and∆(T ,m,)) by ∆.

The significance of the above construction is that the results of [32] show thaquasiprimitive permutation group of type HS, HC, SD, or CD is, for someT , m and), permutationally isomorphic to a subgroup ofW on Ω that contains the socle ofW .Conversely, not all subgroups ofW that contain SocW are quasiprimitive, but we can givnecessary and sufficient conditions for this in terms of the conjugation action on theTof simple direct factors of SocW . For convenience, for eachi = 1, . . . , n set

Ti =(t1, . . . , tn) ∈ SocW : tj = idT for all j = i

so thatT = T1, . . . , Tn and the minimal non-singleton blocks for the action ofW on Tare

T1, T)+1, . . . , T)(m−1)+1, T2, T)+2, . . . , . . . , T), . . . , Tm). (3-A)

Lemma 3.3. Suppose thatG is a subgroup ofW = W(T,m,)) that containsSocW . ThenG is quasiprimitive onΩ if and only if one of the following holds:

(i) G acts transitively onT ;(ii) m = 2, there are precisely twoG-orbits onT , namelyΓ1 andΓ2, and we have∣∣Γi ∩ Tj , Tj+)

∣∣ = 1 for all i = 1,2 andj = 1, . . . , ).

Furthermore,G is primitive on Ω if and only if either (i) above holds and the seT),T2), . . . , Tm) is a minimal non-singleton block for the action ofG onT , or (ii) aboveholds.

If G is quasiprimitive onΩ , thenG is of typeSD or HS if ) = 1, and typeCD or HC if) > 1, depending respectively on whetherG is transitive onT or not.

Proof. This follows directly from [25] and [32].


f

s

n theup,a

an

t

We now suppose thatM is a subgroup ofW = W(T,m,)) that is regular onΩ and is anormal subgroup of SocW . ThusM is isomorphic toT (m−1)) and is the direct product o(m − 1)) minimal normal subgroups of SocW , of which preciselym − 1 lie in any non-trivial block of imprimitivity for the action ofW on the setT of minimal normal subgroupof SocW (cf. (3-A)). Notice that any two such subgroups are conjugate inW . Clearly thenormalizerNW(M) containsM as a regular normal subgroup and so, as observed ifinal paragraph of Section 2.3,NW(M) is permutationally isomorphic to some subgroH say, of HolM in its base group action onM. The following construction and lemmgive an explicit description ofH .

Construction 3.4. As above, letM be any normal subgroup of SocW that is regular onΩ . Recall thatΩ = Ω(T,m,)) is the set of right cosets of∆ = ∆(T ,m,)) in W . Define abijectionφ :Ω →M by

φ :∆m →m for all m ∈M.

Note that the regularity ofM means thatφ is well-defined. Note also that there existsisomorphism from the base groupM of HolM to its centralizer in HolM given by

m → m = m−1mθ for all m ∈M,

wheremθ ∈ AutM is the inner automorphism induced on conjugating bym. For x ∈CSocW(M) we definex ∈ M andx ∈ CHolM(M) by

x = (φ(∆x)

)−1 and x = (x)−1(x)θ .

SetC = CSocW(M), and defineC M andC CHolM(M) in the obvious way. (Note thawe setx = (φ(∆x))−1, rather thanx = φ(∆x), so thatx → x gives rise to an isomorphismC → C.)

Example 3.5. We calculateC andC given thatM is given by

M = (t1, . . . , tm)) ∈ SocW : ti = idT for all i = 1, . . . , )

.

Clearly

C = (t1, . . . , tm)) ∈ SocW : ti = idT for all i = )+ 1, . . . ,m)

∼= T ).

If x = (t1, . . . , tl , id, . . . , id) ∈C, then∆x = ∆m, where for 1 i l, 1 j m − 1, the(i + j l)-entry ofm is t−1

i , andm1 = · · · = ml = 1. Thenx = m−1 and it follows that

C = (t1, . . . , tm)) ∈M: ti = tj if i > ), j > ) andi ≡ j (mod))

=

(id, . . . , id︸︷︷︸) copies

, t1, t2, . . . , t), t1, . . . , t), . . . , t1, . . . , t)): t1, . . . , t) ∈ T

∼= T ).


ge

ting

f,ned

sed to

oup.of

Lemma 3.6. Suppose thatW = W(T,m,)), and let M, φ, C, C, and C be as inConstruction3.4. Then the subgroupNW(M) of the permutation groupW on Ω ispermutationally isomorphic to the subgroupNHolM(C ) of HolM in its base group actionon M. More precisely, ifιφ : Sym(Ω) → Sym(M) is the isomorphism induced byφ, thenιφ(NW(M)) = NHolM(C ).

Proof. We start by noting that the definitions ofφ andιφ ensure that the imageιφ(NW (M))

is contained in HolM, where we view HolM as a subgroup of Sym(M) via its basegroup action onM (cf. Section 2.3). We now claim that ifx ∈ C, then ιφ(x) = x. Tosee this choosem ∈M and consider the images of∆m ∈ Ω underx followed byφ, and ofm = φ(∆m) underx. Firstly, sincex centralizesM,

φ(∆mx)= φ(∆xm) = φ(∆(x)−1m

) = (x)−1m;

secondly, since the action of HolM on M is such thatM HolM acts by rightmultiplication and such that AutM HolM acts by conjugation, we have that the imaof m underx = (x)−1(x)θ is(

mx−1)(xθ ) = x−1(mx−1)x = x−1m.

The two images are equal, and the claim follows.Thus ιφ(C) = C. As NW(M) normalisesC = CSocW(M), we certainly have tha

ιφ(NW(M)) NHolM(C ). We leave the reader to show that equality holds by showthat bothNW(M) andNHolM(C ) are isomorphic to

M.((AutT × Sm−1) S)

).

This is best done by considering the explicit example given above;NW(M) can then becalculated directly, whilst to determineNHolM(C ) we can observe that since HolM =M AutM we have

NHolM(C

) = MNAutM(C

) = MNAutM(C )

,

and then calculateNAutM(C ) directly. The relevance of the above lemma is that ifG is a quasiprimitive subgroup o

NHolM(C ), thenG is permutationally isomorphic to a subgroupG of W ; consequentlywe have the inclusion(G,W). This idea forms the basis of 3-inclusions which are defiin the next section.

We close this section with a discussion of how twisted wreath products can be uconstruct quasiprimitive permutation groups of type ASreg or TW.

Lemma 3.7. Let T twrφ P be the twisted wreath product of a non-abelian simple grT by P with respect toφ; let Q be the twisting subgroup and letBφ be the base groupSuppose thatφ−1(InnT ) is a core-free subgroup ofP . Then the base group action


up

up of

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nite)

nups

isus

s

T twrφ P onBφ is faithful and quasiprimitive; moreover, it is of typeASreg if P = Q, andof typeTW otherwise. If, in addition,φ(Q) InnT , and there does not exist a subgroQ of P strictly containingQ with a homomorphismφ : Q → AutT that extendsφ, thenQ is a core-free subgroup ofP and the permutation groupT twrφ P onBφ is primitive oftypeTW.

Conversely, up to permutational isomorphism, any quasiprimitive permutation grotypeASreg or TW arises in this way.

Proof. This is a straightforward collection of various pieces of information, as availafor instance, in [1]. We remark that the sufficiency of the condition involvingφ(Q) InnT

follows from the ‘Schreier’ conjecture, and hence from the classification of the (fisimple groups.

Suppose now thatT is a non-abelian simple group, thatφ−1(InnT ) is a core-freesubgroup ofP , and thatQ < P so thatT twrφ P on Bφ is a quasiprimitive permutatiogroup of type TW. We wish to identify some naturally occurring quasiprimitive subgroof T twrφ P that will form the basis of 2-inclusions. LetR be a subgroup ofP such thatP = QR, and letη :Q∩R → AutT be obtained by restrictingφ.

Lemma 3.8. The base group action ofT twrη R on Bη is faithful and quasiprimitive;moreover, the permutation groupT twrη P on Bη is of typeTW, and is permutationallyisomorphic to the subgroupBφR of T twrφ P onBφ .

Proof. The core ofη−1(InnT ) in R is⋂r∈R

(η−1(InnT )

)r ⋂r∈R

(φ−1(InnT )

)r =⋂p∈P

(φ−1(InnT )

)psinceP = QR andQ normalizesφ−1(InnT ). However, the last term is trivial as itprecisely the core inP of φ−1(InnT ). The first statement now follows from the previolemma.

To see the ‘moreover’ statement, observe firstly thatT twrη R on Bη is quasiprimitiveof type TW as the twisting subgroupQ∩R is a proper subgroup ofR sinceP = QR andsinceQ is a proper subgroup ofP . Recall from Section 2.4 thatBη comprises all mapf :R → T that satisfy

f (rq)= f (r)η(q) for all r ∈R andq ∈Q∩R.

Likewise,Bφ comprises all mapsf :P → T that satisfy

f (pq) = f (p)φ(q) for all p ∈ P andq ∈Q.

We define a mapBη → Bφ by

f → f for all f ∈ Bη,


ducts.

the

oup

ntext

of

ce

wheref :P → T is given by

f (rq)= f (r)φ(q) for all r ∈R, q ∈Q.

Given that P = QR, we leave it to the reader to verify thatf → f is a well-defined bijectionBη → Bφ , and that it induces an isomorphismχ : Sym(Bη) → Sym(Bφ)

satisfyingχ(T twrη R) = BφR T twrφ P . We must stress that the quasiprimitive permutation groups of type ASreg or TW are not

the only quasiprimitive permutation groups that can be written as twisted wreath proIndeed, ifM is a non-abelian characteristically simple group, then HolM and, in fact, anysubgroup of HolM that containsM as a minimal normal subgroup, can be written astwisted wreath productT twrφ P of a simple direct factorT of M by some top groupPsuch that kerφ is a core-free subgroup ofP [1, 2.5 and 2.7(3)] and such that the base grof the holomorph is identified with the base group of the twisted wreath product.

The following result provides the converse and also information relevant in the coof Lemma 3.6.

Lemma 3.9. Let T twrφ P be a twisted wreath product such thatT is a non-abeliansimple group, andkerφ is a core-free subgroup ofP . For eachp ∈ P let p denote theautomorphism ofBφ induced on conjugation byp. Then the mapT twrφ P → HolBφ

given by

fp → f p for all f ∈Bφ, p ∈ P,

is a monomorphism and induces an embedding of the permutation groupT twrφ P in itsbase group action onBφ into the permutation groupHolBφ in its base group action onBφ .

Let X be the image ofT twrφ P under this map. Note thatCHolBφ (Bφ) ∼= Bφ and sois the direct product of copies ofT . Then there exists a subgroupK of CHolBφ (Bφ) thatis the direct product of disjoint full strips ofCHolBφ (Bφ) with X NHolBφ (K) if and onlyif there exists a subgroupQ of P containing the twisting groupQ with a homomorphismφ : Q→ AutT that extendsφ.

Proof. We start by claiming thatCT twrφP (Bφ) = CoreP (kerφ): this follows from directcalculation, or we may use [1, 2.7(3)]. Thus the mapP → AutBφ given byp → p for allp ∈ P is a monomorphism, and the first statement of the lemma follows.

This leaves the final statement. Given thatBφ obviously normalizes any subgroupCHolBφ (Bφ) and as the action ofP on CHolBφ (Bφ) is equivalent to its action onBφ , wereduce to proving the statement that there exists a subgroupK of Bφ that is the directproduct of disjoint full strips ofBφ and that is normalised byP , if and only if there existsa subgroupQ of P containing the twisting groupQ with a homomorphismφ : Q → AutTthat extendsφ. This assertion follows from [2]. In fact, slightly more follows: we deduthat if K does exist thenK = B

φ Bφ for some appropriateQ andφ.


w-up

s 1–5 areimple

n

ational

itivefor

nup

4. The quasiprimitive-primitive inclusions

Recall that a quasiprimitive-primitive inclusion is a pair(G,H) such thatG, H aresubgroups of Sym(Ω) for some setΩ with G H , G quasiprimitive andH primitive.This section describes ‘inclusions of types 1–5’ and also what is meant by a ‘bloinclusion’, thus clarifying the significance of the entries of the Results Matrix.

We remark that blow-up inclusions and quasiprimitive-primitive inclusions of type3 are entirely natural, whereas quasiprimitive-primitive inclusions of types 4 anddependent on some highly restrictive properties of factorisations of non-abelian sgroups.

4.1. Quasiprimitive-primitive inclusions of type 1

Quasiprimitive-primitive inclusions of type1 are (SD, SD)-inclusions and for such ainclusion(G,H) the groupsG,H have the same socle.

Let G Sym(Ω) be a quasiprimitive permutation group of type SD; thus there is anon-abelian simple groupT and an integerm> 1 such that SocG ∼= T m. By Lemma 3.3and the comments immediately preceding it, we may assume that up to permutisomorphismG is a subgroup ofW = W(T,m,1) onΩ = Ω(T,m,1) such thatG containsSocW ∼= T m and acts transitively by conjugation on them simple direct factors of SocW .It also follows from Lemma 3.3 that an overgroupH of G in W is primitive if and only ifH acts primitively by conjugation on the simple direct factors of SocW ; moreover, suchan overgroup, if primitive, is of type SD.

Quasiprimitive-primitive inclusions isomorphic to such(SD,SD)-inclusions(G,H) aresaid to bequasiprimitive-primitive inclusions of type1, or more simply, 1-inclusions.

We remark that 1-inclusions are the appropriate generalisation for quasiprimgroups of the(SD,SD)-primitive inclusions as described by 3.8 of [31]. Note thatany imprimitive quasiprimitive groupG W = W(T,m,1) of type SD, there exists aprimitive groupH such that(G,H) is a 1-inclusion: we can always takeH = W asW isalways primitive. However the converse need not hold: for example, if the subgroupH ofW(T,3,1) is given by

H = (t1, t2, t3)π ∈W(T,3,1): t1, t2, t3 ∈ T , π ∈ A3

∼= T A3,

thenH is a primitive permutation group of type SD, but, as SocW = T 3 is the only propersubgroup ofH that contains SocW and as SocW is far from quasiprimitive,H containsno proper subgroupG such that(G,H) is a 1-inclusion.


Quasiprimitive-primitive inclusions of type2 are (TW, TW)-inclusions and for such ainclusion(G,H), the groupsG,H have the same socle. For a given quasiprimitive groG of typeTW, necessary and sufficient conditions for the existence of an overgroupH suchthat (G,H) is a 2-inclusion are given in Proposition4.1.


.7t

ys fort this

e

e

2-

s

nd

that

e

Let H Sym(Ω) be a primitive permutation group of type TW. By Lemma 3we may, up to permutational isomorphism, assume thatH is the twisted wreath producT twrφ P acting on its base groupBφ = Ω where the twisting subgroupQ is a core-freesubgroup ofP . Suppose thatR is a subgroup ofP such thatP = QR; let η :Q ∩ R →AutT be the restriction ofφ to Q∩R. Then Lemma 3.8 shows thatG = T twrη R onBη

is a quasiprimitive permutation group of type TW and, moreover, allows us to identifyGwith the subgroupBφR of T twrφ P onBφ . Thus we obtain a(TW,TW)-inclusion (whichis proper if and only ifR is a proper subgroup ofP ).

Quasiprimitive-primitive inclusions isomorphic to such(TW,TW)-inclusions(G,H)

are said to bequasiprimitive-primitive inclusions of type2, or more simply, 2-inclusions.We remark that the quasiprimitive permutation groupG as described above, ma

be imprimitive or primitive: see Lemma 3.7 for necessary and sufficient conditionprimitivity. Indeed, a variety of different types of behaviour can occur and we exhibiwith some examples.

Example 1. T = A5, P = A6, Q = A5, andφ :Q → AutT the natural map; then thcandidates forR distinct fromP are the proper subgroups ofP that are transitive in thenatural action ofP on 6 points, and all of these give rise to 2-inclusions(G,H) with G

imprimitive.

Example 2. T = A5, P = S6, Q = S5, andφ :Q → AutT the natural map; then thcandidates forR distinct fromP are the proper subgroups ofP that are transitive in thenatural action ofP on 6 points, and all but one of these give rise to 2-inclusions(G,H)

with G imprimitive—the exception isR = A6, and in this case we obtain a primitiveinclusion.

Example 3. T = A5, P = M11, Q ∼= S5, andφ :Q → AutT the natural map; thenQ is amaximal subgroup ofP , and the candidates forR distinct fromP are the proper subgroupof P that are transitive on the 66 blocks of the Steiner system preserved byP (see [7]).Using [26] (or information on permutation characters of M11 as printed in [7]) we seethat no such subgroups exist, and soH has no proper subgroupG such that(G,H) is a2-inclusion.

Example 4. T = A6, P = AutM22, Q = NP (M10) ∼= AutA6, andφ :Q → AutT thenatural map; then the candidates forR distinct fromP are the proper subgroups ofP suchthatP = QR. Using [26] (or information on permutation characters of M22 as printed in[7]) we see that the only such subgroup is the socle ofP ; this gives rise to a primitive2-inclusion.

We also remark that not all quasiprimitive permutation groupsG Sn of type TW

possess a proper overgroupH in Sn such that(G,H) is a 2-inclusion. To find necessary asufficient conditions onG for a suitable overgroup to exist, suppose thatG = T twrη R withtwisting subgroupQη is a quasiprimitive permutation group in its base group action,is T is a non-abelian simple group and kerη is a core-free subgroup ofR; further supposethatH = T twrφ P with twisting subgroupQ is a primitive permutation group in its bas


allhe

trs

up

theph

p

e

group action, and moreover, thatR < P , P = QR, Qη = Q∩R, andη = φ|Q∩R, whence(G,H) is a proper 2-inclusion. AsH is primitive, Lemma 3.7 shows thatQ is a core-freesubgroup ofP . This, together with the conditionP = QR, implies thatQη = Q ∩ R iscore-free inR (cf. the beginning of the proof of Lemma 3.8)—as this does not hold inquasiprimitive permutation groups of type TW (see Remark 2.1(a) of [32]) we have tfollowing necessary condition onG:

Qη is a core-free subgroup ofR.

Conversely we have the following.

Proposition 4.1. Suppose thatG = T twrη R with twisting subgroupQη is a quasiprimitivepermutation group in its base group action, and suppose further thatQη is a core-freesubgroup ofR. Setk = |R : Qη| and use the action ofR by right multiplication on the seof right cosets ofQη in R to identifyR with a subgroup ofSk so that the point-stabilizeR1 is equal toQη . (This action ofR is faithful sinceCoreR Qη is trivial.) Then there exista primitive permutation groupH such that(G,H) is a 2-inclusion if and only if thereexists a subgroupP of Sk containingR with a homomorphismφ :P1 → AutT for whichthe following conditions all hold:

(i) φ|R1 = η;(ii) φ(P1) InnT ;(iii) there does not exist a subgroupQ of P strictly containingP1 with a homomorphism

φ : Q→ AutT such thatφ|P1 = φ.

Proof. Other than to note that condition (i) enables the construction of an overgroH

of G, whilst conditions (ii) and (iii) then ensure thatH is primitive so that(G,H) is aquasiprimitive-primitive inclusion, we leave the proof to the reader.

We close our discussion of 2-inclusions by remarking that 2-inclusions areappropriate generalisation of(TW,TW)-inclusions as described by the final paragraof 3.6 of [31].


Quasiprimitive-primitive inclusions(G,H) of type3 have one of the types(TW,HC),

(TW,CD), (ASreg,HS), (ASreg,SD). A summary is given in Subcases1–4 below. In allcases we haveSocG = T l(m−1), SocH = T lm for somel 1, m 2 and non-abeliansimple groupT .

Let G Sym(Ω) be a quasiprimitive permutation group of type TW or ASreg; byLemma 3.7 we may assume that up to permutational isomorphismG is the twisted wreathproductT twrφ P acting on its base groupBφ = Ω whereT is a non-abelian simple grouandφ−1(InnT ) is a core-free subgroup ofP . Suppose that there exist a subgroupQ of Pand a homomorphismφ : Q → AutT such thatQ contains the twisting subgroupQ andφ extendsφ; here we allow the possibility thatQ = Q andφ = φ. We consider the bas


pe

wh

t

hons

p

se, the

,

groupBφ

of the twisted wreath productT twrφP . Now B

φis naturally a subgroup ofBφ ,

and as such is normalised byP . Let ) = |P : Q| and(m − 1) = |Q : Q|. We claim thatthere exists a permutational isomorphism betweenG and a subgroup ofW = W(T,m,))

on Ω(T,m,)). To see this we start by noting that, asBφ is a regular normal subgrouof G, the base group action ofG on Bφ is permutationally isomorphic to that of somsubgroup of HolBφ on Bφ , and we identifyG with this subgroup of HolBφ . Now letMbe any normal subgroup of SocW that is regular onΩ and letC, C, andC be defined asin Construction 3.4. We observe that there exists an isomorphismχ :Bφ → M that mapsBφ

to C: indeed, bothBφ andM are isomorphic to the direct product of)(m − 1) copies

of T , whilst bothBφ

and C are the direct product of) disjoint full strips ofBφ andM

respectively, with each full strip coveringm− 1 simple direct factors. Our aim is to shothatG is permutationally isomorphic to a subgroup ofNW(M). By Lemma 3.6 it is enougto show thatG is permutationally isomorphic to a subgroup ofNHolM(C ) acting on thebase groupM of HolM. In fact, given the above isomorphismχ , it is enough to show thaG, as a subgroup of HolBφ , normalises the subgroupK of HolBφ given by

K = x−1xθ : x ∈B

φ

CHolBφ (Bφ),

wherexθ is the inner automorphism ofBφ induced on conjugating byx ∈ Bφ

Bφ . AsP

normalisesBφ

it is clear thatP normalisesK; asBφ centralisesK we see thatG = BφP

must also normaliseK. The claim now follows.Let H be any primitive overgroup ofG in W(T,m,)) that contains SocW so thatH is

primitive of type HS, HC, SD, or CD.Quasiprimitive-primitive inclusions isomorphic to such inclusions(G,H) are said to be

quasiprimitive-primitive inclusions of type3, or more simply, 3-inclusions.There are various subcases of 3-inclusions(G,H) that are worth distinguishing.

Subcase1. G is of type ASreg and so is necessarily imprimitive. HereQ is equal toP , whenceQ = Q, m = 2 and) = 1. ThusH W(T,2,1) and (G,H) is either an(ASreg,HS)- or an (ASreg,SD)-inclusion. Note that the subgroupG(SocW(T,2,1)) ofW(T,2,1) is primitive of type HS, whilstW(T,2,1) is primitive of type SD. Thus sucinclusions always exist, and the(ASreg,SD)-inclusions can be considered as compositiof (ASreg,HS)-inclusions with a primitive(HS,SD)-inclusion.

Subcase2. G is of type TW andQ = Q. Herem = 2 and) > 1. ThusH W(T,2, ))and (G,H) is either a(TW,HC)- or a (TW,CD)-inclusion. Note that the subgrouG(SocW(T,2, ))) of W(T,2, )) is primitive of type HC, whilstW(T,2, )) is primitiveof type CD. Thus such inclusions always exist, and, in analogy to the above subca(TW,CD)-inclusions can be considered as compositions of(TW,HC)-inclusions with aprimitive (HC,CD)-inclusion.

Subcase3. G is of type TW andQ = P . Herem > 2 and) = 1. Also by Lemma 3.7G is necessarily imprimitive asφ is a proper extension ofφ. We haveH contained inW(T,m,1) and(G,H) is a(TW,SD)-inclusion. Again, asW(T,m,1) is itself primitive,such quasiprimitive-primitive inclusions always exist.


-

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n

6the

ifll

e

arkser

Subcase4. G is of type TW and Q < Q < P . Here m > 2 and ) > 1. Againby Lemma 3.7,G is necessarily imprimitive. We haveH W(T,m,)) and (G,H)

is a (TW,CD)-inclusion. Again, asW(T,m,)) is itself primitive, such quasiprimitiveprimitive inclusions always exist.

We remark that the primitive 3-inclusions, which necessarily belong to Subcase 2 aare precisely the(TW,HC)- and (TW,CD)-inclusions described in the first paragraof 3.6 of [31].


Quasiprimitive-primitive inclusions(G,H) of type 4 are (PA,SD)-inclusions withSocG = T 2, SocH = S2, where the pair(T ,S) is (A5,A6), (M11,M12), or (;7(q),

P;+8 (q)) (q 2). These inclusions are described explicitly below.Let T , S be non-abelian simple groups such thatT S, and letσ be an automorphism

of S such that

S = T(T σ

)and T σ2 = T . (4-A)

In the terminology of [3], the factorisationS = T (T σ ) is a full factorisationof S, and wededuce from Theorem 1.1 of [3] that one of the following holds:

(i) T = A5, S = A6, andσ is any automorphism ofS that is not induced on conjugatioby an element ofS6 and such thatσ 2 normalisesT .

(ii) T = M11, S = M12 andσ is any outer automorphism ofS such thatσ 2 normalisesT .(iii) T = ;7(q), S = P;+

8 (q), whereq = pf (p prime), andσ is any automorphism ofSnot in InnS·(Zf × Z3) if q is even, and not in InnS·(Zf × A4) if q is odd, and inadditionσ 2 normalisesT butσ does not.

To see this deduction we note firstly that the only full factorisationsS = AB in which S,A andB are all non-abelian simple groups andA ∼= B correspond to lines 1, 2, 5 andof Table 1 of [3]; of these, line 1 corresponds to the first possibility above, line 2 tosecond, while for line 6 we haveT = Sp6(2) ∼= ;7(2) andS = ;+

8 (2) so case (iii) holdswith q = 2. Finally suppose that line 5 of [3, Table 1] holds. ThenT = ;7(q), S = P;+

8 (q)

with q > 2. Here AutS induces a transitive permutation group on the setT of S-conjugacyclasses of subgroups isomorphic toT , and this permutation group isS3 of degree 3 ifqis even, and isS4 of degree 6 (in its action on unordered pairs from a set of size 4)q

is odd. In both cases the kernel of this action is InnS·Zf as field automorphisms leave aclasses inT fixed setwise. Ifq is even thenS = AB with A ∼= B ∼= T if and only if A,B

lie in different classes inT ; so forq even case (iii) holds. Ifq is odd the action of AutSonT is imprimitive with 3 blocks of size 2, and we haveS = AB with A ∼= B ∼= T if andonly if theS-conjugacy classes containingA,B lie in different blocks of size 2; the setwisstabiliser inS4 of the pair ofS-conjugacy classes containingA andB is therefore cyclic oforder 2 generated by a transposition, so again case (iii) holds.

Conversely, suppose that (i), (ii) or (iii) holds. By Table 1 of [3], and by the remabove in case (iii), we see that (4-A) is indeed satisfied. LetL denote the setwise stabilis


)

(ii)

ter

s

w

for

,

in AutS of the pair ofS-conjugacy classes containingT andT σ . Then in cases (i) and (iiwe haveL = AutS; while in case (iii) we haveL = InnS · (Zf × Z2) andσ /∈ InnS · Zf .In all cases the following two conditions hold.

σ ∈L \ InnS, σ 2 ∈ InnS, (4-B)

and

CL(T ) = CL

(T σ

) = id. (4-C)

In the following, we make little reference to the explicit possibilities given by (i),and (iii), and instead assume only thatT andS are non-abelian simple groups withT < S

andσ ∈ AutS such that conditions (4-A), (4-B) and (4-C) all hold, whereL is the setwisestabiliser in AutS of the pair ofS-conjugacy classes containingT andT σ .

Consider the primitive permutation groupW = W(S,2,1) on the setΩ = Ω(S,2,1)of type SD as defined by Construction 3.2. Note that|Ω | = |S|. Recall thatΩ is the setof right cosets of∆ = ∆(S,2,1) in W ; thus∆ is a point inΩ . To distinguish the poin∆ from the subgroup∆ we writeω = ∆ ∈ Ω for the former. Note that the point-stabilizWω is equal to∆. Define

M = (t1, t

σ2

): ti ∈ T

= T × T σ ∼= T 2

and by identifyingS with InnS view M as a subgroup of SocW = S2. As

Mω = M ∩∆ = (t, t): t ∈ T ∩ T σ

we have that

|M : Mω| = |T |2|T ∩ T σ | = ∣∣T (

T σ)∣∣ = |S|

and soM is a transitive subgroup ofW . Now (σ,σ )(1 2) is an element ofW , and since(t1, t

σ2

)(σ,σ )(1 2) = (tσ

2

2 , tσ1)

andσ 2 ∈ NAutS(T ) by (4-A), it follows that(σ,σ )(1 2) normalisesM and interchangethe simple direct factors ofM on conjugation. SinceM is transitive onΩ , we haveNW(M) = MNWω(M). Now (ϕ,ϕ) ∈ NWω(M) if and only if ϕ ∈ NAutS(T )∩NAutS(T

σ ),and (ϕ,ϕ)(1 2) ∈ NWω(M) if and only if ϕ interchangesT and T σ . It follows thatNW(M) (s1, s2)π ∈ W(S,2,1): s1, s2 ∈ L. A direct computation using (4-C) noshows thatCW(M) is trivial. Hence if the subgroupG of NW(M) containsM andinterchanges the simple direct factors ofM on conjugation (and such subgroups exist,exampleNW(M) is a possibility), thenM is the unique minimal normal subgroup ofG,whenceG is quasiprimitive onΩ asM is transitive onΩ ; moreover,G is quasiprimitive oftype PA sinceMω is a subdirect product of(T ∩ T σ )2 M. LetG be any such subgroup


asirectle is a

ly up

s of

or

n

gf

and chooseH to be any subgroup ofW containingG(SocW). Then asG is necessarilytransitive on the simple direct factors of SocW , Lemma 3.3 implies thatH is primitive onΩ of type SD. AlsoG is imprimitive onΩ (by the O’Nan–Scott Theorem, see [25])if the point-stabilizer in the socle of a primitive permutation group is a proper subdsubgroup of some proper subgroup of the socle, then the point-stabilizer in the socsubdirect subgroup of the socle; however inG we have forω ∈ Ω

(SocG)ω = Mω <(T ∩ T σ

)2< T × T σ = M = SocG

with Mω subdirect in(T ∩ T σ )2, but withMω not subdirect inM.

Quasiprimitive-primitive inclusions isomorphic to such(PA,SD)-inclusions(G,H) aresaid to bequasiprimitive-primitive inclusions of type4, or more simply, 4-inclusions.

We remark that quasiprimitive-primitive inclusions of type 4 may be described explicitto isomorphism. To see this we suppose that(S,T ,σ ) and(S, T , σ ) both satisfy conditions(4-A), (4-B), and (4-C); set

M = (t1, t

σ2

): ti ∈ T

and M = (

t1, tσ2

): ti ∈ T

,

and let I (S,T ,σ ), I (S, T , σ ) be the sets of 4-inclusions as defined above in term(S,T ,σ ), (S, T , σ ) respectively. Thus(G,H) ∈ I (S,T ,σ ) if and only if

M G G(SocW) H W = W(S,2,1)

andM is the unique minimal normal subgroup ofG; a similar statement also holds felements inI (S, T , σ ). We make the following sequence of claims.

(1) If (G,H) ∈ I (S,T ,σ ) and (G, H ) ∈ I (S, T , σ ) are isomorphic inclusions, theS ∼= S.

(2) If S ∼= S and(G, H ) ∈ I (S, T , σ ), then there exists(G,H) ∈ I (S,T ,σ ) with (G,H)

isomorphic to(G, H ).(3) Suppose that(G,H) and(G, H ) are both inI (S,T ,σ ). Then(G,H) is isomorphic

to (G, H ) if and only if G = G andH = H .(4) If S = A6 and (G,H) ∈ I (S,T ,σ ), then there are two possibilities forG with

G/M ∼= 2, one possibility forG with G/M ∼= 22, and the entries in the followinmatrix give the number of possibilities forH givenG, with the isomorphism type oG/M as indicated by the row label and the isomorphism type ofH/SocW by thecolumn label:

( 2 22 23

2 1 3 122 0 1 1

).

Thus|I (S,T ,σ )| = 12.


e

w

s

)s

(5) If S = M12 and (G,H) ∈ I (S,T ,σ ), then there is a unique possibility forG (withG/M ∼= 2), and there are two possibilities forH , namelyH = G(SocW) andH = W .Thus|I (S,T ,σ )| = 2.

(6) If S = P;+8 (q) with q = pf (p prime), then the number of possibilities for(G,H)

depends onf . GivenT andσ , the groupNW(M)/M ∼= OutT × S2, andG is anysubgroup satisfyingM G NW(M) such thatG/M projects non-trivially onto thedirect factorS2; H is any subgroup ofW containingG(SocW).

To see (1) observe that the isomorphism between(G,H) and (G, H ) induces anisomorphism between SocH and SocH . Now SocH = SocW(S,2,1) ∼= S2 and SocH =SocW(S,2,1) ∼= S 2, and the claim follows.

To see (2) we start by noting that an arbitrary isomorphism betweenS andS can be usedto construct a permutational isomorphism fromW(S,2,1) to W(S,2,1). By replacingG,H by their images under this permutational isomorphism, we may assume thatS = S.From now on we writeW for W(S,2,1). If one of the assertions (i), (ii) or (iii) madafter (4-A) holds, then AutS is transitive on pairs(A,B) of S-classesA,B ∈ T such thatT = AB for A ∈ A, B ∈ B. Thus there existsx ∈ AutS such that(T )x is S-conjugate toT and(T σ )x is S-conjugate toT σ . Clearly we may choosex such that(T )x = T . Now theconditionS = T T σ implies thatT acts transitively by conjugation on theS-class ofT σ ,so we may further assume that(T σ )x = T σ . Therefore the element(x, x) ∈ W conjugatesM to M and hence conjugates(G, H ) to an inclusion inI (S,T ,σ ), and claim (2) follows.

We turn to (3). Certainly(G,H) is isomorphic to(G, H ) if G = G and H = H .Thus it is enough to assume that(G,H) and (G, H ) are isomorphic and then to shothatG = G andH = H . So there exists a permutational isomorphismα betweenH andH which by restriction gives a permutational isomorphism betweenG andG, and alsotherefore restricts to permutational isomorphisms between the socles ofH andH , that isbetweenS ×S and itself, and between the socles ofG andG, that is betweenM and itself.A permutational isomorphism fromS × S to itself corresponds to an automorphismα ofS ×S that preserves the(S ×S)-conjugacy class of the point-stabilizer(S ×S)ω, where asaboveω ∈Ω is such that

(S × S)ω = (S × S) ∩∆ = (s, s): t ∈ S

.

Similarly asα induces a permutational isomorphism betweenG andG, α restricts to anautomorphism ofM that preserves theM-conjugacy class of the point-stabilizerMω.

By replacingα by its product with some inner automorphism ofS × S induced by anelement ofM, if necessary, we may suppose thatα is an automorphism ofS × S thatnormalisesM and (S × S)ω. A straightforward calculation shows thatα ∈ Aut(S × S)

normalisesM and(S × S)ω if and only if either (a)α = (α0, α0) for someα0 ∈ AutS thatnormalises bothT andT σ , or (b)α = (α1, α1)(1,2) for someα1 ∈ AutS that interchangeT and T σ . In either caseα ∈ NW(M). SinceGω contains an element ofNW(M) thatinterchanges the two simple direct factors ofM, we may assume thatα is as in (a). Wenote that the element of OutS corresponding to the automorphismα0 occurring in case (alies in the centre of OutS (for in cases (i) and (ii), OutS is abelian, while in case (iii) thiassertion follows from the discussion preceding (4-B)).


mefal

in

n

lds:

Thus the action ofα on M is equivalent to that induced on conjugation by soelement ofNW(M) ∩ NW(Mω). Given thatCW(M) is trivial, whence any element oNW(M) is determined uniquely by its action onM, we deduce that the permutationisomorphism betweenG andG is in fact one induced by conjugation withinNW(M). Thusthe isomorphism between(G,H) and(G, H ) is also one induced by conjugation withNW(M). SinceW/SocW ∼= OutS × S2, and sinceα = (α0, α0) with α0 in the centreof OutS, conjugation byα fixes any subgroup ofW containing SocW , and henceH =(H )α = H . Similarly sinceNW(M)/M ∼= OutT × S2 is abelian in all cases, conjugatiobyα fixes any subgroup ofNW(M) containingM, and henceG = (G )α = G. In particular,conjugation byα preserves the pairG,H) ∈ I (S,T ,σ ), and claim (3) follows.

Verification of (4), (5) and (6) is by direct calculation and is left to the reader.


Quasiprimitive-primitive inclusions(G,H) of type 5 are (AS,PA)-inclusions withSocG = T , SocH = S2, and |Ω | = d2, where(T ,S, d) are as in Table1. They are thegeneralisations to the quasiprimitive case of the primitive(AS,PA)-inclusions listed in[31, Table 3] (but see Remark4.4).

Let T be a non-abelian simple group with proper subgroupsA andB such that

T = AB and |A| = |B|. (4-D)

In the terminology of [3], the factorisationT = AB is a full factorisation ofT , and wededuce from Theorem 1.1 of [3] that (4-D) holds if and only if one of the following ho

(i) T = A6, A ∼= A5, andB = Aα for some automorphismα of T that is not induced onconjugation by an element ofS6;

(ii) T = M12, A ∼= M11, andB = Aα for some outer automorphismα of T ;(iii) T = Sp4(q) with q > 2, q even,A ∼= L2(q

2).2, andB = Aα for some automorphismα ∈ AutT \ (T NAutT (A));

Table 1(T ,S, d) for 5-inclusions

T S d

A6 A6 6

M12 T 12A12

Sp4(q), q even,q > 2 Tq2(q2−1)

2Ad

Sp4r (q0), q = qr0

P;+8 (q) T

q3(q4−1)(2,q−1)

Ad

Sp8(2), q = 2 120


the

f [3]

plicitp

t

f

t

vel

(iv) T = P;+8 (q), A∼= ;7(q), andB = Aα for some ‘triality’ α ∈ AutT (here a triality is

an outer automorphism ofT of order 3 corresponding to a symmetry of order 3 ofDynkin diagram of typeD4).

(The above four possibilities correspond to lines 1, 2, 4 and 5, 6 of Table 1 orespectively.) We observe that in all cases there existsα ∈ AutT such that

B = Aα, (4-E)

and furthermore, thatA, B are both maximal subgroups ofT . In the following definition ofquasiprimitive-primitive inclusions of type 5, we make no further reference to the expossibilities given by (i)–(iv), and instead assume only thatT is a non-abelian simple grouwith α ∈ AutT and maximal subgroupsA, B such that (4-D) and (4-E) both hold.

Let Γ be the set of right cosets ofA in T : we view T as a permutation group onΓvia the right multiplication action ofT . Let Ω = Γ × Γ and view the wreath producSym(Γ ) S2 as a permutation group onΩ via the action given by

(γ1, γ2)(x1, x2)π = (γ1πx1π, γ2πx2π)

for all γ1, γ2 ∈ Γ , x1, x2 ∈ Sym(Γ ) andπ ∈ S2. (Note that this is an action sinceπ = π−1

for all π ∈ S2.) Define the subgroupX of Sym(Γ ) S2 by

X = (t, tα

): t ∈ T

Sym(Γ )× Sym(Γ ). (4-F)

Observe that the stabilizer inX of the point(A,A) ∈ Ω is the subgroup(t, tα

): t ∈ T

∩ (A×A) = (t, tα

): t ∈A∩Aα−1 ∼= A∩Aα;

asT = AB, we see thatA∩B = A∩Aα has index|Γ |2 = |Ω | in T , whenceX is transitiveonΩ .

We claim thatCSym(Γ )S2(X) is trivial. As T is a primitive non-regular subgroup oSym(Γ ), its centralizer in Sym(Γ ) is trivial; asX is a full diagonal subgroup ofT 2, wesee that the centralizer in Sym(Γ )2 of X is also trivial. So, ifx ∈ Sym(Γ ) S2 is non-trivialand centralisesX, thenx /∈ Sym(Γ )2 and we have

x = (y, z)(1 2) for somey, z ∈ Sym(Γ ).

As x2 ∈ CSym(Γ )2(X), we havex2 = id whencez = y−1. For t ∈ T , we deduce by direccalculation thatx centralises(t, tα) ∈ X if and only if tα = ty . HenceAα equalsAy , andso is an intransitive subgroup of Sym(Γ ) sinceA is a point-stabilizer inT in its action onΓ . However, the factorisationT = AAα implies thatAα is a transitive subgroup ofT —acontradiction.

LetG be any subgroup of Sym(Γ ) S2 that containsX as a normal subgroup. The aboclaim, together with the fact thatX ∼= T is simple, implies thatX is the unique minimanormal subgroup ofG, whence the transitivity ofX implies thatG is quasiprimitive of typeAS. The next lemma gives a means of constructing primitive inclusions starting fromG.


so

f

hats

h

l

Lemma 4.2. LetH1 be any overgroup ofT in Sym(Γ ), let S = SocH1, and letH be anysubgroup ofSym(Ω) that containsS × S as a minimal normal subgroup, and that alcontainsG. Then(T ,H1) is a primitive inclusion onΓ and S is a non-abelian simplegroup containingT ; H is contained inSym(Γ ) S2, and moreover, is primitive of typePAwith SocH = S2; and(G,H) is an(AS,PA)-inclusion.

Quasiprimitive-primitive inclusions isomorphic to such(AS,PA)-inclusions(G,H) aresaid to bequasiprimitive-primitive inclusions of type5, or more simply, 5-inclusions.

Proof. We assume the notation of the lemma and suppose thatH is any subgroup oSym(Ω) containingS × S as a minimal normal subgroup; we must show thatH iscontained in Sym(Γ ) S2 and is primitive of type PA with SocH = S2. Recall thatSis the socle ofH1, whereH1 is an overgroup ofT in Sym(Γ ). As A is maximal inT

it follows that T is primitive onΓ so the inclusion(T ,H1) is primitive and hence isdescribed by [31]. Given thatT is non-abelian and simple, the results of [31] show teitherH1 is almost simple withT SocH1 = S, orT ∼= L2(7). The latter is impossible aone of (i)–(iv) above holds, and soS is a non-abelian simple subgroup of Sym(Γ ) whichcontainsT , whenceS is also primitive. In particular,S is primitive and non-regular asTis primitive and non-regular. We deduce that the centralizer ofS in Sym(Γ ) is trivial. AlsoasS is transitive onΓ , we see that the orbits of the minimal normal subgroups ofS ×S onΩ = Γ 2 are of the form

(γ, δ): γ ∈ Γ

or(δ, γ ): γ ∈ Γ

,

whereδ is some element ofΓ . As H normalisesS × S, H preserves the set of sucorbits, whenceH Sym(Γ ) S2 since the latter is the largest subgroup of Sym(Ω) topreserve the set of such orbits. We observed above thatCSym(Γ )(S) = id; it follows thatthe centralizer ofS × S in Sym(Γ ) S2, and so also inH , is also trivial. HenceS × S isthe unique minimal normal subgroup ofH . Now the point-stabilizer inS × S of the point(γ, γ ) ∈Ω is preciselySγ ×Sγ ; asS is primitive onΓ and asH interchanges the minimanormal subgroups ofS, it is straightforward to see thatSγ × Sγ is a maximalH -invariantproper subgroup ofS × S. The primitivity of H follows. As it is visible thatH is of typePA, this finishes the proof.

Next we show that the simple groupsT ,S and the sized = |Γ | (which depends only onT and not onS) for a 5-inclusion must satisfy one of the lines of Table 1.

Lemma 4.3. Let T S Sym(Γ ) be as in Lemma4.2, that is bothT and S are non-abelian simple groups andΓ is the set of right cosets inT of a subgroupA of T such thatthere existsα ∈ AutT with T = A(Aα). Suppose thatL satisfies

T L Sym(Γ ) and L ∼= S.

ThenL= S, andT ,S andd := |Γ | are as in one of the lines of Table1.


that

;

tof

ed bynsn)

notesated

ited.

ps,

Proof. By Lemma 4.2,(T ,S) is a primitive inclusion with bothS andT non-abelian andsimple, and moreoverT satisfies one of cases (i)–(iv) above. The results of [24] showone of the following holds:

(a) S = T or S = Alt (Γ );(b) case (iii) holds and the inclusion(T ,S) is derived from line 2 of Table VI of [24]

that is,T = Sp4(q) with q even andq > 2, S = Sp4r (q0) with q = qr0, andΓ can be

identified with the set of right cosets ofO−4r (q0) in S so that|Γ | = 1

2q2(q2 − 1) and a

point-stabilizer inT is O−4 (q) ∼= L2(q

2).2;(c) case (iv) holds and the inclusion(T ,S) is derived from line 5 of Table VI of [24]; tha

is, T = P;+8 (2), S = Sp8(2), andΓ can be identified with the set of right cosets

O−8 (2) in S so that|Γ | = 120 and a point-stabilizer inT is ;7(2) ∼= Sp6(2).

(We have used the phrase ‘is derived from’ for two reasons: (i) not all inclusions listthe relevant line of Table VI of [24] apply, and (ii) Table VI of [24] only lists inclusio(G,H) with G maximal inH , whereasT need not be maximal inS.) The second assertionow follows, and it remains to prove thatL = S. If (a) holds, then this is immediate. If (bor (c) hold, then we leave the reader to verify that the following all hold:

(1) S is Sym(Γ )-conjugate toL (equivalently, that the point-stabilizer inS is determinedup to AutS-conjugacy by knowledge ofT and the isomorphism type ofS);

(2) NSym(Γ )(S) is transitive on the Sym(Γ )-conjugates ofT contained inS;(3) NSym(Γ )(T ) NSym(Γ )(S).

Given (1) and (2) an easy argument shows thatS is NSym(Γ )(T )-conjugate toL, whence(3) implies thatL= S as required. Remark 4.4. The second author has realised that there is an error in theaccompanying Table 3 of [31]. Note (a) fails to take into account the possibility indicby (b) in the above proof, and should be amended as follows:

Note (a) for Table 3 of [31]:In all casesSocL0 is embedded inSocL1 as a diagonal subgroup ofS ×S, andSocL1is S × S, or Am ×Am, or line 3 applies withq = qr

0, r > 1, andSocL1 is PSp4r (q0).

Lemmas 4.2 and 4.3 imply that the possibilities for 5-inclusions are severely limTo give a precise meaning to this remark we suppose thatG, H are subgroups of Sym(Ω)

such that(G,H) is a 5-inclusion. It follows from the definition that SocG is non-abelianand simple, that SocH is the direct product of two isomorphic non-abelian simple grouand that (

SocG,NSym(Ω)(SocH))

is also a 5-inclusion. We claim that


e-to-

alh

lep

e theit

by

byble 3s the

y

up to isomorphism of inclusions, the possibilities for this latter inclusion are in onone correspondence with the possible isomorphism types ofSocG andSocH .

To verify the claim we assume thatT is a non-abelian simple group with a maximsubgroupA and an automorphismα ∈ AutT such that (4-D) and (4-E) both hold witB = Aα ; we letΓ andX be as defined above in terms of the tuple(T ,A,α), see (4-F), andfrom now on identifyT with the subgroup of Sym(Γ ) induced by the right multiplicationaction ofT on Γ ; we let S be any subgroup of Sym(Γ ) that is non-abelian and simpand that containsT , and setH = NSym(Γ )(S) S2, which we view as a permutation grouon Γ × Γ in the natural way. We further assume thatT , A, α, Γ , X, S, andH satisfyanalogous conditions. Note that both

(X,H) and(X, H

)are 5-inclusions; note also that the above claim can now be rephrased as:

The5-inclusions(X,H) and(X, H ) are isomorphic if and only ifT ∼= T andS ∼= S.

This is immediate in one direction, and so it is enough to assume thatT ∼= T andS ∼= S,and to show that the two 5-inclusions are isomorphic. It is clear that we may usisomorphism betweenT andT to assume thatT = T . On inspecting each of the explicpossibilities given by (i)–(iv) above, we see thatA is AutT -conjugate toA, whence thereexists a permutational isomorphism between the permutation groupsT on Γ andT onΓ , whereΓ , Γ are respectively the sets of right cosets ofA, A in T . It follows that wemay assume thatA = A andΓ = Γ . On again inspecting the explicit possibilities given(i)–(iv) above, we see that there existβ,γ ∈ T NAutT (A) such that

α = βαγ.

As β,γ ∈ T NAutT (A) there existx, y ∈ NSym(Γ )(T ) such that conjugation byx, y inducesthe same automorphism ofT asβ , γ respectively. Direct calculation shows that

X(x−1,y) = X,

where (x−1, y) is considered as an element of Sym(Γ ) × Sym(Γ ). Thus (X, H ) isisomorphic to the inclusion(X, H (x,y−1)). By Lemma 4.3,S = S = S x = S y−1

; asH (x,y−1) is the normalizer in Sym(Γ ) S2 of S x × S y−1

, it must equalH and the claimholds.

We finish our consideration of 5-inclusions by remarking that if(G,H) is a 5-inclusionwith G primitive, then one of cases (i)–(iii) applies, that is SocG is isomorphic toA6,M12 or Sp4(q) with q even andq > 2, and moreover, the primitive inclusion is listedTable 3 of [31] (as modified by Remark 4.4). Conversely, all inclusions listed by Taof [31] are 5-inclusions. Thus 5-inclusions are perhaps most profitably thought of ageneralisation to the quasiprimitive case of the primitive(AS,PA)-inclusions described bTable 3 of [31].


vácsoups,

oduct

se

s

-ceptdefine

s.

t

p

po-

4.6. Blow-ups and quasiprimitive-primitive inclusions

The concept of a blow-up of primitive permutation groups was introduced by Koin [20]. We discuss a generalisation of this concept for quasiprimitive permutation grand give in the last paragraph of this subsection a definition of a blow-up-inclusion.

In the course of defining 5-inclusions we saw our first example of a wreath practing on a Cartesian power, namely Sym(Γ ) S2 on Γ 2. More generally ifK is anypermutation group on a setΓ and) is a positive integer, then the wreath productK S)can be viewed as a permutation group onΓ ) via the action in which the action of the bagroupK) is given by

(γ1, . . . , γ))(x1, . . . , x)) = (γ1x1, . . . , γ)x)) for all γ1, . . . , γ) ∈ Γ,

x1, . . . , x) ∈K, and the action of the top groupS) is given by

(γ1, . . . , γ))y−1 = (γ1y, . . . , γ)y) for all γ1, . . . , γ) ∈ Γ andy ∈ S).

This action is commonly referred to as theproduct actionof the wreath product. (Jameand Kerber [15] call this permutation group the ‘exponentiation’ ofK by S).) In theterminology of Kovács [20] the permutation groupK S) onΓ ) is an example of a ‘blowup’ of the permutation groupK onΓ . The present paper requires both the blow-up conand a weakening of it: we start by defining the weaker concept, and then go on tothe blow-up concept in more general terms than in [20].

The product action ofK S) on Γ ) may be viewed in a different way as followA Cartesian decompositionE of index) is a setE = Γ1, . . . ,Γ) of ) partitionsΓi ofΩ such that, for all choices ofγi ∈ Γi (1 i )) the intersection

⋂i γi is a singleton.

Thus there is a well-defined map

Θ :∏i

Γi → Ω,

and it is easy to check thatΘ is a bijection. Suppose then thatE is a Cartesiandecomposition ofΩ . We will usually identifyΩ with Γ1 × · · · × Γ), and we say thaE is non-trivial if ) > 1. A permutationg ∈ Sym(Ω) is said toleaveE invariant if, for alli ), Γ g

i ∈ E . The set of all permutations ofΩ which leaveE invariant forms a subgrouof Sym(Ω) and is called thestabiliserof E in Sym(Ω). If |Γ1| = · · · = |Γ)| then we saythatE is homogeneous, and in this case we may identifyΩ with Γ ), whereΓ = Γ1, andthe stabiliser ofE is then Sym(Γ ) S) acting in the product action.

Now suppose thatG Sym(Ω) leaves a non-trivial homogeneous Cartesian decomsitionE of index) invariant, soΩ = Γ ) andG Sym(Γ ) S). Before giving the remainingdefinitions we need the following notation. SetX = Sym(Γ ) S), and letπ :X → S) bethe projection map of the wreath productX onto its top groupS) so that kerπ is the basegroup (Sym(Γ ))) of X. For i = 1, . . . , ), let Xi be the inverse image inX underπ ofthe stabilizer inS) of the pointi; thusXi

∼= Sym(Γ ) × (Sym(Γ ) S)−1). For eachi, letπi :Xi → Sym(Γ ) be the natural projection map.


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p

roups

n-

tnedsion

3-

espe-t to

The Cartesian decompositionE is said to betransitive forG, orG-transitive, if π(G) isa transitive subgroup ofS); it is G-intransitiveotherwise. Thecomponentsfor G relativeto E are the permutation groupsGi = πi(Xi ∩ G) Sym(Γ ) for i = 1, . . . , ). If E is aG-invariant Cartesian decomposition that isG-transitive, thenE is homogeneous and thcomponents are necessarily permutationally isomorphic to each other; in such a ccan (by [20, 2.2]), and always will, conjugate by an element ofX so that the Cartesiadecomposition is such thatG is contained in the subgroupG1 S) of X, and such that alits components are equal toG1. Finally aG-invariant Cartesian decompositionE is saidto be a blow-up decompositionof Ω relative toG if E is G-transitive andG contains thedirect product of the socles of its components, that is

G SocG1 × · · · × SocG).

(Note that SocG1 × · · · × SocG) is naturally a subgroup of(Sym(Γ ))) X.) In such asituation the permutation groupG onΩ is said to be ablow-upof the permutation grouG1 onΓ .

The significance of the above concepts in the context of primitive permutation gis made clear by the following results.

Proposition 4.5. Suppose thatG on Ω is a primitive permutation group that leaves ivariant a non-trivial homogeneous Cartesian decompositionE of index) with componentsG1, . . . ,G). Then the following all hold:

(i) E is G-transitive;(ii) the components are all primitive;(iii) if G is not a blow-up of its componentG1, then the primitive inclusion(G,G1 S))

is either an(AS,PA)-inclusion listed in Table3 of [31] (but see also Remark4.4ofthe present paper—these are primitive5-inclusions), or is a (TW,HC)-inclusion andis described in the first paragraph of3.6of [31] (these are primitive3-inclusions).

Proof. Parts (i) and (ii) are easy consequences of the primitivity ofG (cf. line 6, p. 307of [20]). To see (iii) we observe that for primitive permutation groupsG the present concepof “G being a blow-up” coincides, up to permutational isomorphism, with that defiin [20] (see also [31]), and we use the results of [31] to analyse the primitive inclu(G,G1 S)).

The inclusions in Proposition 4.5(iii) are the primitive 5-inclusions and primitiveinclusions described earlier in this section.

Theorem 4.6. If G is a blow-up of the primitive groupG1 and the socle ofG1 is notregular, thenG is primitive.

Proof. This is part of Theorem 1 of [20]. Cartesian decompositions left invariant by quasiprimitive permutation groups,

cially blow-up decompositions, are studied in [4]. More precisely, in [4] we attemp


-

ain-

de-. Ifs

icate), (4)

ts

e

cs

re the

ss

describe all quasiprimitive permutation groupsG that leave invariant a non-trivial Cartesian decompositionE with; (i) E intransitive forG; (ii) E transitive forG, but not a blow-updecomposition; and (iii)E a blow-up decomposition. If (i) holds, then we succeed in obting an explicit description of all suchG, and in particular we find that anyG-intransitiveCartesian decomposition invariant under a quasiprimitive permutation groupG involvesprecisely twoG-orbits on the partitions. If (iii) holds, then we find that we are able toscribe essentially all suchG as blow-ups of smaller quasiprimitive permutation groups(ii) holds, then we are able to deduce a great deal aboutG, but the information obtained inot as explicit as in the other two cases.

We extract from the results obtained in [4] the following theorem; parts (1), (2) indthat the blow-up concept behaves well with respect to quasiprimitivity, and parts (3are needed in the proof of Theorem 1.2.

Theorem 4.7. Let G on Ω be a transitive permutation group, and letE be a non-trivialhomogeneousG-invariant Cartesian decomposition ofΩ of index ) such thatG hascomponentsG1, . . . ,G) Sym(Γ ). Then the following all hold:

(1) If E is a blow-up decomposition andG is quasiprimitive onΩ , then the componenG1, . . . ,G) are all quasiprimitive onΓ .

(2) If E is a blow-up decomposition and the componentG1 is quasiprimitive onΓ and notof typeHA, thenSocG = SocG1 × · · · × SocG) andG is quasiprimitive onΩ .

(3) G is not quasiprimitive of typeSD.(4) If G is quasiprimitive of typeCD, then E is a blow-up decomposition and th

componentG1 is quasiprimitive of typeSD or CD.

Remark 4.8. Let G be a quasiprimitive permutation group. IfG has a non-trivial blow-updecomposition, then it follows from parts (1) and (2) of the above theorem that SoG isnot simple, and also that a point-stabilizer in SocG is either trivial or is not simple; thuG is not of type HS, nor type AS, nor type SD. Conversely, ifG is of type HC or CD, thenG is a non-trivial blow-up. This is well-known forG primitive; if G is imprimitive and soof type CD, then it is straightforward to see thatG is the blow-up of groups of type SD,and possibly also of type CD, in an exactly analogous fashion to the case forG of typeCD. Moreover, it follows from the proof of Lemma 8.2 below that these analogues aonly blow-up decompositions possessed by groups of type CD. On the other hand, ifG isof type HA, TW, or PA, thenG may or may not be a non-trivial blow-up.

We shall also need the following lemma about Cartesian decompositions.

Lemma 4.9. LetG onΩ be a permutation group, and letE be a non-trivial homogeneouG-invariant Cartesian decomposition ofΩ of index ) such thatG has componentG1, . . . ,G) Sym(Γ ). As usual identifyG with a subgroup ofSym(Γ ) S), and letπ1, . . . , π) be the projection maps used to define the componentsG1, . . . ,G). Let M bea transitive non-abelian minimal normal subgroup ofG with M (Sym(Γ ))). Then thefollowing both hold:


,

e

l

l

mapsisds, andf

(1) If |πi(M)| |Γ |2 for all i = 1, . . . , ) with equality in each if and only ifM is notregular, then

M = π1(M)× · · · × π)(M) (Sym(Γ )

)). (4-G)

(2) If (4-G)holds thenπ1(M) is a transitive minimal normal subgroup ofG1 with π1(M)

regular if and only ifM is regular; alsoE is G-transitive. Furthermore, if, in additionCSym(Γ )(π1(M)) = id, then bothG andG1 are quasiprimitive andE is a blow-updecomposition ofΩ relative toG with componentG1.

Proof. Let T be a minimal normal subgroup ofM so thatT is a non-abelian simplgroup andM is the direct product of theG-conjugates ofT ; thus M ∼= T k wherek = |G : NG(T )|. For convenience, set

M = π1(M)× · · · × π)(M). (4-H)

NowM, and so alsoT , is certainly contained inM . Also for i = 1, . . . , ), the imageπi(M)

is a homomorphic image ofM, whenceπi(M) ∼= T ki for some integerki , 0 ki k, andwe have

M ∼= T (k1+···+k)).

(HereT 0 denotes the trivial group.) By viewingM as the direct product of its minimanormal subgroups, each of which is isomorphic toT , we see that the subgroupT ofM is either a minimal normal subgroup ofM, or is a non-trivial full strip ofM . As G

normalisesM , we deduce that anyG-conjugate ofT is also either a minimal normasubgroup ofM , or is a non-trivial full strip ofM , depending on the nature ofT . If theformer, that isT and itsG-conjugates are minimal normal subgroups ofM, thenM isa normal subgroup equal to the direct product of its images under the projectionM → πi(M); it follows from (4-H) thatM = M as required in (1). So to prove (1) itenough to reach a contradiction under the assumption that the hypothesis of (1) holthat eachG-conjugate ofT is a non-trivial full strip ofM . ThusM is the direct product onon-trivial full strips ofM. These full strips must be disjoint as they commute inM, andso

k1 + · · · + k) 2k,

or equivalently,|M| |M|2. However the transitivity ofM implies that

|M|2 |Ω |2 = |Γ |2),

with equality if and only ifM is regular, whilst the hypothesis of (1) implies that

|M|2 ∣∣M∣∣ = )∏∣∣πi(M)

∣∣ |Γ |2),
i=1


e

rect

f

oupalso

the

with the latter inequality being an equality if and only ifM is not regular. This gives threquired contradiction.

We turn to (2): we continue with the above notation and assume thatM = M . Thetransitivity of M on Ω implies thatπi(M) is a transitive subgroup of Sym(Γ ) fori = 1, . . . , ); in particular, eachπi(M) is non-trivial. Also the normality ofM in G impliesthatπi(M) is a normal subgroup ofGi for i = 1, . . . , ). AsG is contained in Sym(Γ ) S),we see thatG permutes the direct factorsπ1(M), . . . , π)(M) of M on conjugation. Thusthe fact thatM is minimal normal inG together with the above observation that each difactorπi(M) is non-trivial, forcesG to act transitively on the direct factors (equivalentlyEisG-transitive) and forcesG1 to act transitively on the simple direct factors ofπ1(M). Thusπ1(M) is a transitive minimal normal subgroup ofG1 as required. Also the transitivity oG on the direct factorsπ1(M), . . . , π)(M) of M means thatπ1(M) is regular onΓ if andonly if each ofπ1(M), . . . , π)(M) is regular onΓ ; as the latter is equivalent toM regularonΩ , we see thatπ1(M) is regular onΓ if and only if M is regular onΩ .

We assume further thatCSym(Γ )(π1(M)) is trivial. This implies thatπ1(M) is theunique minimal normal subgroup ofG1, whence SocG1 = π1(M) andG1 is certainlyquasiprimitive asπ1(M) is transitive and contained in every non-trivial normal subgrof G1. We have shown thatE is aG-transitive Cartesian decomposition and hence wehave

CSym(Γ )

(πi(M)

) = id for all i = 1, . . . , ),

and we can similarly deduce thatπi(M) = SocGi for i = 1, . . . , ). ThusG containsthe direct product of the socles of its components, namelyM, and soE is a blow-updecomposition relative toG. Finally direct calculation shows that

CSym(Γ )S)(M) =)∏

i=1

CSym(Γ )

(πi(M)

) = id,

whenceM is the unique minimal normal subgroup ofG, andG is quasiprimitive for thesame reason thatG1 is quasiprimitive.

We close this section with a definition of blow-up inclusions, thus completingexplanation of the notation used in the Results Matrix.

Definition 4.10. Let G H be permutation groups onΩ , and letE be a non-trivialhomogeneousH -invariant Cartesian decomposition ofΩ of index ). ThenE is alsoG-invariant. LetG1, H1 be components ofG, H respectively (so thatG1 H1 Sym(Γ )).If E is a blow-up decomposition relative to bothG andH , and if both(G,H) and(G1,H1)

are quasiprimitive-primitive inclusions, then we say that(G,H) is ablow-up inclusion, andis ablow-upof (G1,H1).

Remark 4.11. It follows from the proof of Lemma 6.8 given below that all(CD,CD)-inclusions are blow-up inclusions of(SD,SD)-inclusions, and all(CD,PA)-inclusions areblow-up inclusions of either(SD,AS)-inclusions or(CD,AS)-inclusions.


na

hereple

hen

oup

uppose,

zes

-

s of

it

5. The proof of Theorem 1.4

Suppose thatH is an almost simple group with simple socleS, and thatH = AB

whereA is a proper subgroup ofH not containingS, andB ∼= T k for some non-abeliasimple groupT and integerk 2. By the ‘Schreier’ conjecture (the truth of which isconsequence of the finite simple group classification)H/S is soluble and thereforeB S.ThusS = (A∩S)B, and so, to prove Theorem 1.4, it is sufficient to consider the case wH = S. We therefore assume thatH = S is simple. It is a consequence of the finite simgroup classification thatS contains a cyclic Sylowp-subgroup, for some prime divisorpof |S|. The following simple lemma for such primes will be useful.

Lemma 5.1. Suppose thatS = AB is a proper factorisation of a simple groupS such thatB = T k for some non-abelian simple groupT andk 2. If a primep dividing |S| is suchthat the Sylowp-subgroups ofS are cyclic, then thep-part |S|p of |S| divides|A|, andp

does not divide|B|.

Proof. Suppose thatp divides|S : A| = |B : B ∩ A|, so thatp divides|B| = |T |k . Thenp divides|T | and henceB contains a subgroupZk

p , contradicting the assumption that tSylow p-subgroups are cyclic. Similarly ifp divides|B|, then we obtain a contradictioby the same argument.

Now we consider the various possibilities forS. Let A, B be maximal subgroups ofScontainingA,B respectively, so thatS = AB = AB.

Lemma 5.2. If S is an alternating group, a sporadic simple group, or an exceptional grof Lie type, then Theorem1.4 is true.

Proof. Suppose first thatS = An with n 5. SinceS T × T we must haven 10. IfA ∼= An−1 then Theorem 1.4 holds, so we may assume that this is not the case. Snext thatAn−r A An ∩ (Sn−r × Sr) for somer = 1, . . . ,5. Then by our assumptionr 2. SinceS = AB, the groupB acts transitively onr-element subsets of a set of sin, and this is impossible by [5, 5.3] and [16], sinceB = T k . ThusA does not have thiform, and it follows from [26, Theorem D] thatAn−r B = T k Sn−r × Sr for somer = 1, . . . ,5, andA acts transitively onr-element subsets of a set of sizen. However inthis case we would needr = n/2 = 5, k = 2, andA would need to be transitive on 5element subsets and there is no proper subgroup ofA10 with this property.

Next suppose thatS is an exceptional group of Lie type. The proper factorisationS are listed in [26, Table 5], and in none of them is one of the factors a product ofk 2simple groupsT .

Finally suppose thatS is a sporadic simple group. SinceS has a proper factorisation,is one of the groupsL of [26, Table 6], and in particularS = McL,Co2,Co3. Supposefirst that S = J2 or He. Then for each of the primesp listed in column 2 of the linecorresponding toS of the top part of [28, Table 10.6], a Sylowp-subgroup ofS is cyclicand so by Lemma 5.1,p divides|A|. For the same reason 5 divides|A| whenS = M11 or


of

ftorsa

.2.e

pairs4],

lar

eans,lanest no

nly

Table 2

S A

M11 11 · 5, L2(11)M12 11· 5, L2(11), M11M24 M23HS M22

M12. It follows from [28, Theorem 4] thatA is one of the subgroups listed in column 3[28, Table 10.6]. We deduce thatS,A satisfy one of the lines of Table 2.

However, on checking the list of maximal factorisationsS = AB with A containing asubgroupA of this form, we see that in none of these cases doesB contain a subgroup othe formT × T . It remains to considerS = J2 and He. In these cases none of the facA, B occurring in a maximal factorisationS = AB as listed in [26, Table 6] containssubgroupT × T .

Thus we may assume thatS is not isomorphic to one of the simple groups in Lemma 5This means thatS is a classical simple group defined on ann-dimensional vector spacV (n, q) over a field of orderq = pf , wherep is a prime andn 2. For a groupL letRp(L) denote the minimal dimension of a non-trivial projective representation ofL incharacteristicp. First we deal with some of the small parameters and identify possibleS,A for n 5. The information crucial for this lemma is provided by [28, Theoremwhich relies heavily on the finite simple group classification.

Lemma 5.3. The dimensionn 4, the pair(n, q) = (4,2) or (4,3), and ifn 5, thenS,Aare as in one of the lines of Table3.

Proof. Now Rp(S) Rp(B) 2k 4 (see [17, Proposition 5.5.7]), and in particun 4. SinceT × T B < S, pk divides |S| for some primep 5, and it follows that(n, q) = (4,2), (5,2) or (4,3), and if (n, q) = (6,2) then S = L6(2) andB = L3(2)2.Also, if S = ;+

8 (2) thenB = A25.

If S = L6(2) andB = L3(2)2, then|A| is divisible by 31· 5, which implies thatA P1or P5 (parabolic subgroups; see, for example, [30, Theorem 3.1] and [10]). This msinceS = AB, that the groupB must be transitive on the 1-spaces and on the hyperpof V (6,2). However it follows from a theorem of Hering, see [23, Appendix 1], thasubgroup of the formT k (k > 1) has this property. Thus(n, q) = (6,2).

If S = ;+8 (2), then the only maximal subgroup containingB = A2

5 is B = (A5 × A5) :22 (see [7, p. 85]). Also, sinceS = AB and using [26, Table 4], we see that the o

Table 3PossibleS,A for classicalS of dimension at least 5

S A Conditions

PSp2m(q) or P;2m+1(q) A ;−2m(q) m 4,m even

P;+2m(q) A ;2m−1(q) m 4,m even

(m,q) = (4,2)


yor

s

eover,a

nes

in

as

).up

of

lane)]).s

.on

),

maximal subgroup containingA is A ∼= ;7(2) ∼= Sp6(2) and moreover, applying a trialitautomorphism if necessary, we may assume thatA is the stabiliser of a non-singular vectv ∈ V (8,2) and B is the stabiliser;+

4 (4) · 22 of an extension field structure onV :=V (8,2). ThusB = ;+

4 (4) and, sinceS = AB, B is transitive on the|S : A| = 120 non-singular vectors inV . As in [26, 3.6.1(c)], we see thatB preserves a non-singular GF(4)-quadratic formP onV and a vectorw ∈ V is non-singular if and only ifP(w) /∈ GF(2). Itfollows sinceB preservesP thatB leaves invariant the 60 vectorsw with P(w) = P(v),and therefore thatB is intransitive on the non-singular vectors inV , a contradiction. ThuS = ;+

8 (2).Next we apply [28, Theorem 4]. All the remaining groupsS of dimensionn 5, apart

from those in the setX := Sp8(2),L7(2),;−8 (2),;−

10(2),;+10(2), occur in column 1 of

[28, Table 10.1] and are not one of the exceptions listed in column 5 of that table. Morfor S /∈ X, and for each of the primesr occurring in column 2 of [28, Table 10.1],Sylow r-subgroup ofS is cyclic and so, by Lemma 5.1,r divides|A|. It follows from [28,Theorem 4] thatS,A are as in one of the lines of Table 3. Now suppose thatS ∈ X. Ananalogous argument applies toS, r in [28, Table 10.4] and we deduce that one of the liof Table 3 holds.

Next we deal with the groups occurring in line 2 of Table 3.

Lemma 5.4. S = P;+2m(q).

Proof. Suppose thatS = P;+2m(q). Then by Lemma 5.3,m is even,m 4,(m,q) = (4,2),

andA has a normal subgroup;2m−1(q). Applying a triality automorphism if necessarythe casem = 4, we may assume thatA stabilises a non-singular 1-spaceU . Let A, B bemaximal subgroups ofS containingA,B respectively, so thatS = AB. Then A is thestabiliser ofU . We consider in turn the various maximal factorisations of this formclassified in [26, Theorem A], that is,S = P;+

2m(q) = AB with ‘ A= N1’ andm even. Webegin with the first three lines of [26, Table 1] for the groupS = P;+

2m(q).Subcase1. B is a parabolic subgroupPm or Pm−1, or is the stabiliserGLm(q) · 2/Z of

a pair of maximal totally singular subspaces ofV (2m,q) (whereZ denotes the scalarsHereB = (A∩ B )B, and the quotientC of B modulo its largest soluble normal subgrois almost simple with socle Lm(q). SinceB = T k, it follows thatB is isomorphic to asubgroupC of C, and by the ‘Schreier Conjecture’,C is contained in the simple socleC. SinceA is the stabiliser ofU in S, the subgroupA ∩ B projects onto a subgroupDof C such thatD meets Lm(q) in the stabiliser of a hyperplane, or a 1-space–hyperppair, of the spaceV (m,q) on which Lm(q) acts naturally (see [26, 3.6.1(a) and 3.6.1(bMoreover we have a proper factorisationC = DC. ThusC is transitive on the hyperplaneof V (m,q), and hence also on the 1-spaces ofV (m,q). However it follows from a theoremof Hering, see [23, Appendix 1], that no subgroup of the formT k (k > 1) has this property

Subcase2. B = GUm(q) · 2, preserving a quadratic extension field structureV (2m,q). Here we have a proper factorisationB = (A ∩ B )B and A ∩ B stabilises anon-singular 1-space of the spaceV (m,q2) on whichB acts naturally (see [26, 3.6.1(c


of

of

on

e

t is

f

1–

on,

nd

4],ame

],line is

and 3.5.2(b)]). As in the previous subcase it follows thatB is transitive on the 1-spacesV (m,q2), and we have a contradiction by [23, Appendix 1].

Subcase3. B = PSp2(q) ⊗ PSpm(q), preserving a tensor product decompositionV (2m,q), andq > 2. It was proved in [26, 3.6.1(d)] thatS = AD whereD = PSpm(q) B,and so we have a factorisationD = (A∩D)(B ∩D) with A∩D = PSpm−2(q). It followsthat B ∩ D is transitive on the non-singular 2-spaces of the vector spaceV (m,q) onwhich D acts naturally. Sinceq > 2 it follows from [26, Theorem A] thatm = 6, q iseven, andB ∩ D G2(q). ThusD = (A ∩ D)G2(q) and so|A ∩ G2(q)| = |A ∩ D|/|D :G2(q)| = q(q2 − 1). SinceD is normal inB, we haveB ∩ D normal inB and henceB∩D = T k′

for somek′ k. If B∩D = G2(q) then we have a further proper factorisatiG2(q) = (A ∩ G2(q))(B ∩ D). However, see [26, Theorem B], the only group G2(q)

(q even) which admits a proper factorisation is G2(4) and in this case neither of thfactors has orderq(q2 − 1) = 60. HenceB ∩ D = G2(q) = T k′

, soT = G2(q). HoweverB = PSp2(q)⊗ PSp6(q) has no subgroupB = G2(q)

k with k 2.Subcase4. In order to deal with the remaining relevant line of [26, Table 1], i

convenient first to deal with the unique relevant line of [26, Table 2], namely wherem = 8and B = ;9(q) · a (a 2) acting on the spin module forB. SinceB = B, we have asecond proper factorisationB = (A ∩ B )B and by [26, 4.6.3(a)], the socle ofA ∩ B is;7(q). However, Lemma 5.1 applied to this factorisation implies that|A ∩ B| is divisibleby a primitive prime divisor ofq8 − 1, which is not the case.

Subcase5. We now consider the remaining line of [26, Table 1], namelyq = 2, m 6,and B = ;+

m(4) · 22, preserving a quadratic extension field structure onV (2m,2). HereB = (A∩ B )B, andA∩ B stabilises a non-singular 1-space of the vector spaceV (m,4) onwhich B acts naturally (see [26, 3.6.1(c)]). Thus we have a maximal factorisationB = CD

whereC is the stabiliser of a non-singular 1-space ofV (m,4) andD is a subgroup oB containingB which is maximal subject to not containing;+

m(4). By [26, Theorem A]and [27],D ∩ ;+

m(4) is one ofPm,Pm−1, GLm/2(4) · 2/〈−1〉, GUm/2(4) · 2,PSp2(4) ⊗PSpm/2(4) (if m/2 is even), or;9(4) ·a (if m = 16). By the arguments given in Subcases4 above, each of these factorisations leads to a contradiction.

Subcase6. Next we consider the unique relevant line of [26, Table 3], namelym = 12,q = 2 andB = Co1. Here B = (A ∩ B )B is a proper factorisation and it follows fromLemma 5.1 applied to this factorisation that|A ∩ B| is divisible by 11· 13 · 23. However(see [7]) there is no such proper subgroup of Co1.

Subcase7. Finally we consider the relevant lines from [26, Table 4], so from nowS = P;+

8 (q) andq 3. In lines 1 or 6 of [26, Table 4],B is the image ofA = ;7(q)

under a triality automorphism, orq is a square andB = ;−8 (q1/2) respectively. Here

A ∩ B = G2(q) or G2(q1/2) respectively (see [26, Lemmas A and B on p. 105]) a

B = (A∩ B )B. By Lemma 5.1,|A∩ B| is divisible by a primitive prime divisor ofq4 − 1orp2f −1 (whereq = pf ) respectively, which is a contradiction. In line 2 of [26, TableB is P1,P3 or P4. The latter two possibilities were dealt with in Subcase 1, and the sargument shows thatB = P1 cannot arise. The groupsB in lines 3, 4 and 5 of [26, Table 4were dealt with in Subcases 2, 1 and 3 respectively. The next, and final, relevantline 12 of [26, Table 4] withq = 3 and B = ;+(2). Here A ∩ B = 26 · A7 (see [26,
8


].r

cef

1 of

.ce

le 2]

ione wee

sines 2

Lemma C on p. 106]) andB = (A∩ B )B. By [7], B = A25 and hence|B : 26 ·A7| = 1080

divides|B| = 3600 which is a contradiction. This completes the proof.Now we treat the groups in line 1 of Table 3.

Lemma 5.5. S = P;2m+1(q) or PSp2m(q) with m 3.

Proof. Suppose thatS = PSp2m(q) or that q is odd andS = P;2m+1(q). Then byLemma 5.3,m is even,m 4, andA has a normal subgroup;−

2m(q). Let A, B be maximal

subgroups ofS containingA,B respectively, so thatS = AB andA = NS(A). We considerin turn the various maximal factorisations of this form as classified in [26, Theorem A

Subcase1. We treat together the third line forS = PSp2m(q) and the unique line foS = P;2m+1(q) of [26, Table 1], namely the case whereB = Pm. Here the quotientCof B modulo its largest soluble normal subgroup is almost simple with socle Lm(q), andthe subgroupD of C corresponding toA ∩ B stabilises a hyperplane of the vector spaV (m,q) on whichC acts naturally (see [26, 3.2.4(a) and 3.4.1]). LetC be the subgroup oC corresponding toB. ThenC ∼= B andC = DC, and we deduce thatC acts transitivelyon the hyperplanes ofV (m,q). This is impossible by the argument given in Subcasethe proof of Lemma 5.4.

Subcase2. Next we complete consideration of the orthogonal groupsS = P;2m+1(q).There are only two remaining cases withm even,m 4. The first is line 8 of [26,Table 2] whereS = P;13(3f ) and B = PSp6(3

f ) · a (a 2) acting on its spin moduleHereA ∩ B = Sp2(3

f ) × Sp6(32f ) · 2/Z (see [26, Lemma A on p. 85]). However sin

B = (A ∩ B )B it follows from Lemma 5.1 that|A ∩ B| is divisible by a primitive primedivisor of q12 − 1, which is a contradiction. The second case is line 10 of [26, Tabwhere we haveS = P;25(3f ) andB = F4(3f ). HereB = (A∩ B )B but by [26, Table 5]there are no factorisations ofB of this type.

Subcase3. From now on we assume thatS = PSp2m(q) with m even,m 4. Weconsider first line 6 of [26, Table 1] for PSp2m(q), where we haveq = 2 andB = ;+

2m(2).

HereB = (A∩ B )B andA∩ B = ;2m−1(2) (see [26, 3.2.4(e)]), and this is a contradictby Lemma 5.4. Next we consider the unique relevant line of [26, Table 3] wherhaveS = PSp8(2) and B = S10. Here A ∩ B = S7 × S3 (see [26, 5.1.9]), and we havB = (A ∩ B )B. It follows that B A10 and A10 = (A ∩ A10)B, and this contradictLemma 5.2. Since there are no relevant lines of [26, Table 2], there remain only land 4 of [26, Table 1] to be dealt with.

Subcase4. Consider now line 4 of [26, Table 1] whereq is even,B = Spm(q) S2,andA = O−

2m(q). HereA ∩ B = O−m(q)× O+

m(q) (see [26, 3.2.4(b)]), and henceA∩ B is

contained in the derived groupB ′ = Spm(q)2. ThereforeA∩ B = A∩ B ′ and so

|AB ′| = |A| · |B ′||A∩ B ′| = |A| · |B|

2|A∩ B| = |S|2

which implies thatS = AB ′. However since|B/B ′| = 2 it follows thatB ⊆ B ′ and soS = AB ′. This is a contradiction.


d

at

o

do not

,

].

et

hatthe

e

Subcase5. Thus line 2 of [26, Table 1] holds, that is,q is even andB = Sp2a(qb) · b

for some primeb wherem = ab. Here A ∩ B = O−2a(q

b) · b (see [26, 3.2.1(d)]), an

B ′ = (A ∩ B ′)B. Replaceb by the largest divisor ofm such thatB is contained inSp2a(q

b) · b, wherem = ab, and replaceB by this subgroup. Then we still have thA ∩ B = O−

2a(qb) · b is maximal inB andB ′ = (A∩ B ′)B. Moreover, since Sp2a(q

b) · bcontainsB it follows that 2a 2k 4 (see [17, Proposition 5.5.7]). LetC be a maximalsubgroup ofB ′ containingB, so B ′ = (A ∩ B ′)C is a maximal factorisation, and smust occur in [26, Tables 1, 2 or 3]. It follows from the maximality ofb that B = Pa , orSpa(q

b) S2 (with a even), or G2(qb) (with a = 3). SinceS = AB we haveB ′ = (A∩B ′)B

and the arguments given in Subcases 1 and 4 show that the first two possibilitiesarise, while the third is impossible by Lemma 5.3.

It follows from Lemmas 5.3, 5.4 and 5.5 thatS has dimensionn = 4 andq 4. Tocomplete the proof of Theorem 1.4 we show that these cases give no examples.

Lemma 5.6. The dimensionn of the classical simple groupS is not4.

Proof. Suppose thatS = L4(q),PSp4(q), or U4(q), with q = pf . We will refer to thesecases as caseL, Sp andU respectively. By Lemma 5.3,q 4. By [17, Proposition 5.5.7and Sections 5.3, 5.4], we havek = Rp(T ) = 2 and eitherT = A5, or T = L2(p

e), forsomee dividing f in casesL and Sp, or 2f in caseU, see in particular [17, 5.4.6Thus thep-part of |B| is at mostq2 in casesL andSp, and at mostq4 in caseU. Since|S : B| = |A : A∩B|, it follows that thep-part of |A| is at leastq2 in all cases.

Recall thatqi denotes a primitive prime divisor ofqi − 1, and that such exist for allq ifi 3 except for(q, i)= (2,6). In all cases a primitive prime divisorq4 exists and divides|S|, and a Sylowq4-subgroup ofS is cyclic so, by Lemma 5.1,q4 divides|A|. For the samereasonq3 divides|A| in caseL, andq6 divides|A| in caseU. By [28, Table 10.3], thereare no proper subgroupsA of S with these properties in caseL or caseU. Thus we are incaseSp.

SinceB containsB = T 2, it follows that B is not one of the groupsP1,P2,O−4 (q),

PSp2(q2).2, or Sz(q) (q even). Therefore, from [26], we deduce thatq is even and

(A, B ) = (O−4 (q),L2(q) S2) or (A, B ) = (Sz(q),L2(q) S2), or an image of one of thes

pairs under a graph automorphism ofS. In each of these pairsB is the stabiliser of a direcsum decompositionV (4, q) = U ⊕ W with U,W of dimension 2, andB B ′ = L2(q)

2

leaving each ofU,W invariant. It is proved in [26, 3.2.4(b)] and [26, Lemma, p. 96] tA∩ B ∼= O+

2 (q)× O−2 (q) or D2(q−1) respectively. In the former case it is clear, and in

latter case it follows from the proof of [26, Lemma, p. 96] thatA∩B fixesU andW setwiseand hence is contained inB ′. ThusA∩ B = A∩ B ′, and so|AB ′| = |AB|/2= |S|/2 whichimplies thatS = AB ′, a contradiction. 6. The proof of Theorem 1.2

Throughout this sectionΩ = 1, . . . , n andG H Sn with G quasiprimitive oftypeX andH primitive of typeY so that(G,H) is an(X,Y )-inclusion. Furthermore, w


tionh

rmal

finite

o

t

oup

assume thatG is not of type HA, HS, or HC. (Recall that Theorem 1.2 makes no menof quasiprimitive-primitive inclusions(G,H) with G of type HA, HS, or HC as sucinclusions are necessarily primitive and so have been classified in [31].) LetM = SocGandN = SocH . Let T , S be minimal normal subgroups ofM, N respectively. Letω ∈ Ω .

We start by observing thatM is the unique minimal normal subgroup ofG, andM isnon-abelian; this holds sinceG is quasiprimitive of type AS, TW, SD, CD or PA. Thusthe socleM of G is characteristically simple and non-abelian, whence the minimal nosubgroupT of M is a non-abelian simple group, and we haveM ∼= T k for some integerk 1. In our first lemma we deal with the case whereY = HA. The proof of this lemmarelies on the finite simple group classification since it uses the classification [11] ofsimple groups having subgroups of prime power index.

Lemma 6.1. TheHA column of the Results Matrix(Fig. 1) holds.

Proof. We assume thatH is primitive of type HA: thusH is an affine primitivepermutation group andN is an elementary abelianp-group that is regular onΩ . Hencen= pa for some primep and integera. The form ofn implies thatG is not a quasiprimitivepermutation group of type SD, CD, or TW, which in turn implies that the(X,HA)-entriesof the Results Matrix are correct forX = SD, CD, or TW. There are two entries left tconsider, namely those corresponding toG of either type AS or type PA.

We suppose thatG is of type AS, that is,M = T is a non-abelian simple group. AsT istransitive onΩ we deduce from Corollary 2 of [11] thatT , and hence alsoG, is primitiveonΩ—this justifies the(AS,HA)-entry.

We now suppose thatG is of type PA. Note that to justify the(PA,HA)-entry it is infact sufficient to reach a contradiction under the assumption thatG is imprimitive of typePA. Given the results of [32] we may assume thatM = T k with k > 1, and, for someω ∈ Ω , the point-stabilizerMω is a subdirect product ofRk for some proper subgroupRof T . Note thatGω is transitive on the simple direct factors ofM, sinceG = MGω (by thetransitivity ofM) and sinceM is a minimal normal subgroup ofG. Letσ be the projectionmapM = T k → T k−1 with

kerσ = T × id × · · · × id︸︷︷︸k−1 copies

.

SetK = Mω ∩ kerσ which is a proper subgroup ofT . Then|Mω| = |K||σ(Mω)| and

n = pa = |M : Mω| = |T : K|∣∣T k−1 : σ(Mω)∣∣.

HenceK has prime power index inT . Corollary 2 of [11] shows thatK is a maximalsubgroup ofT whenceK = R and, asGω is transitive on thek direct factors ofT k , wededuce thatMω = Rk whereR is a maximal subgroup ofT . It follows thatMω is a maximalGω-invariant subgroup ofM. AsM is transitive onΩ , we haveG = MGω; we deduce thaGω is a maximal subgroup ofG, whenceG is primitive—a contradiction.

Henceforth we assume thatH is not primitive of type HA. By [25],N (the socle ofH ) is non-abelian and characteristically simple, whence the minimal normal subgrS


pn the

g

ctiont

t

onesosep

n

n nonethe

of N is a non-abelian simple group, and we haveN ∼= S) for some integer) 1. WriteN = S1 × · · · × S), whereS1 = S ∼= Si , and letπi :N → Si be the natural projection ma(1 i )). We now apply results from [3,28], and note that these results also rely ofinite simple group classification. For an almost simple groupL, by aproper factorisationof L we will mean a factorisationL = AB where neitherA norB contains SocL.

Lemma 6.2. LetG, H be as above, that is(G,H) is a quasiprimitive-primitive inclusionwithG quasiprimitive, but not of typeHA, HS, or HC, andH primitive, but not of typeHA.Then the socleM of G is contained in the socleN of H . Moreover one of the followinholds.

(1) T ∼= S andM is a direct product of some simple direct factors ofN ;(2) Y = AS, andN = S has a proper factorisationS = SωM;(3) Y = HS or SD, ) = 2, (T ,S) is (A5,A6), or (M11,M12), or (;7(q), P;+

8 (q)) forsomeq 2, andM = π1(M)× π2(M) ∼= T 2;

(4) Y = HC or CD, H is a blow-up of a primitive group of typeHS or SD respectivelywith socleS2, (T ,S) is as in part(3), andπi(M) ∼= T for eachi;

(5) Y = PA, andS has a proper factorisationS = π1(Nω)π1(M).

Proof. The first paragraph of the proof of Lemma 4.5 of [31] shows that the interseG∩N is non-trivial. SinceM is the unique minimal normal subgroup ofG, we deduce thaM N as required. Suppose thatM contains a simple direct factor ofN , sayS. Then eachG-image ofS is also a simple direct factor ofN contained inM. Without loss of generalitylet S1, . . . , Sk′ be theG-images ofS. ThenS1 × · · · × Sk′ is aG-invariant subgroup ofM.SinceM is a minimal normal subgroup ofG, it follows thatM = S1 ×· · ·×Sk′ andk′ = k,so part (1) holds.

Thus we may assume thatM contains no simple direct factor ofN . SetBi = πi(M) for1 i ), and note that eachBi

∼= T ki for someki , sinceBi is a quotient ofM. By [28,Theorem 2] it follows that either (2) or (5) holds, orY ∈ HS,SD,HC,CD. Moreover ifY = HS or SD then) = 2 or 3, and by [28, Lemma 4.3],Bi < Si for all i. Suppose firsthatY ∈ HS,SD with ) = 2. Then by [28, Lemma 4.5],S = B1A with A ∼= B2. In theterminology of [3], this is a full factorisation ofS, that is,π(S) ⊆ π(B1) ∩ π(A), wherefor a finite groupL, π(L) denotes the set of prime divisors of|L|. All full factorisationsof almost simple groups were determined in [3, Theorem 1.1, Table I], and the onlyfor which both factorsB1 andA are direct powers of the same simple group are thwith (T ,S) as in (3) andπi(M) ∼= T for eachi. Also sinceM is a subdirect subgrouof π1(M) × π2(M) ∼= T × T and M is transitive onΩ of degree|S| it follows thatM = π1(M) × π2(M). Suppose next thatY = HS or SD with ) = 3. Then by [28,Lemma 4.8] there is someL such thatS L AutS andL has a strong triple factorisatiorelative to subgroupsA = NL(B1), B = NL(B2), C = NL(B3) (identifying B2 andB3with subgroups ofS), that is,L = A(B ∩ C) = B(C ∩ A) = C(A ∩ B). All strong triplefactorisations have been classified by the authors in [3, Theorem 1.2, Table V], and iof the examples do the factorsA,B,C have normal subgroups which are powers ofsame simple groupT . Thus ifY = HS or SD then (3) holds. Finally suppose thatY = HCor CD. By [28, Theorem 2],H is a blow-up with componentH0 a primitive group of type


sups

f,

(1)

-

esion

fe

, and

here

re

SD with socleS)′where)′ = 2 or 3. If )′ = 3 then it follows from the proofs of Lemma

8.7–8.9 of [28] thatS has a strong triple factorisation relative to three of the subgroBi , and this is impossible as before. Hence)′ = 2. By [28, Lemma 8.6] and its prooS = B1A with A isomorphic toBi for somei > 1. It follows as above that(T ,S) is as in(3), thatBi

∼= T for eachi, and that the subgroup of SocH0 induced byM is isomorphicto T ∼= T .

We now deal with the various possibilities for the typeY .

Lemma 6.3. TheHS andHC columns of the Results Matrix(Fig. 1) hold.

Proof. We assume thatH is primitive of either type HS or type HC: thusH has preciselytwo minimal normal subgroupsN1 andN2, both isomorphic toS)/2 where) is necessarilyeven, and bothN1 andN2 are regular onΩ . We may takeN1 = S1 × · · · × S)/2 andN2 = S)/2+1 × · · · × S).

By Lemma 6.2,M N and part (1), (3), or (4) holds. Suppose first that partholds. Without loss of generality we may assume thatS1 = T M. SinceM is the directproduct of theG-conjugates ofT , it follows that M N1. Then sinceM is transitiveand N1 is regular onΩ we must haveM = N1. So certainlyM is regular and nonabelian. HenceG is of type ASreg or TW, depending on whetherM is simple or not.By Lemma 3.7 we can writeG as a twisted wreath productT twrφ P with twistinghomomorphismφ :Q → AutT acting on its base groupBφ = Ω , whereφ−1(InnT ) isa core-free subgroup ofP . By settingQ = Q, φ = φ and then following the procedurdescribed whilst defining 3-inclusions, we see that the quasiprimitive-primitive inclu(G,H) is a 3-inclusion; moreover, the inclusion(G,H) belongs to subcase (1) ifH is oftype HS, and to subcase (2) ifH is of type HC.

Suppose next that Lemma 6.2(3) or (4) holds. ThenM is a subdirect product o∏)i=1Bi

∼= T ), and hence by Lemma 2.1,M = ∏kj=1Tj is a direct product of pairwis

disjoint full stripsTj of∏)

i=1Bi . If one of these strips is trivial then, sinceG permutes theTj transitively by conjugation, they are all trivial and we may assume thatT1 = B1 N1.The argument of the previous paragraph then implies thatM = N1, which is not true forcase (3) or (4) since in these casesT ∼= S.

The proof that the AS column of the Results Matrix holds relies on Theorem 1.4hence relies on the finite simple group classification.

Lemma 6.4. TheAS column of the Results Matrix(Fig. 1) holds.

Proof. We assume thatH is primitive of type AS, soN = SocH = S, and thatH =Alt(Ω) or Sym(Ω). Since the Results Matrix makes no claims about the case wX = AS, we assume thatX ∈ TW, SD, CD, PA. SoM = SocG = T k with k > 1, andby Lemma 6.2,M N = S andS = SωM is a proper factorisation. By Theorem 1.4, theare no such factorisations.Lemma 6.5. TheTW column of the Results Matrix(Fig. 1) holds.


on-

t

eringsult in

eter

rye4.5

Proof. We assume thatH is primitive of type TW: thus the socleN of H is regular, non-simple, and non-abelian. By Lemma 6.2,M is a transitive subgroup ofN and soM = N ,whenceG is quasiprimitive of type TW since its socle is regular, non-simple and nabelian. We use Lemma 3.7 to assume that, up to permutational isomorphism,H is thetwisted wreath productT twrφ P acting on its base groupBφ = Ω where the twistingsubgroupQ is a core-free subgroup ofP . Note thatBφ = SocH = N . Let R = G ∩ P .ThenG = BφR and, asBφ = M is also a minimal normal subgroup ofG, we deduce thaR is transitive on the simple direct factors ofBφ ; equivalently, thatRQ = P . It is nowclear that the quasiprimitive-primitive inclusion(G,H) is a 2-inclusion.

We turn to the justification of the SD column. For later convenience when considthe CD column, we deduce the required result as a corollary of a more general rewhich we consider subgroups of the primitive permutation groupH that contain a transitiveminimal normal subgroup.

Lemma 6.6. Let H be a primitive permutation group of typeSD with socleN = S).Suppose that the subgroupG of H has a minimal normal subgroupM that is transitive.Then one of the following holds:

(i) M = N ;(ii) M is a maximal normal subgroup ofN ;(iii) up to permutational isomorphism,H is a subgroup ofW(S,2,1) with S = A6, M12,

or P;+8 (q) (q 2), the subgroupM is the unique minimal normal subgroup ofG

(whenceG is quasiprimitive), and(G, H ) is a 4-inclusion.

Proof. We start by fixing some notation. We identifyN with S), and fori = 1, . . . , ) set

Si =(s1, . . . , s)) ∈ S): sj = idS for all j = i

S) = N;

note that theSi are the simple direct factors ofN . As H is primitive of type SD, thepermutation groupH is permutationally isomorphic to a subgroup ofW(S, ),1): thismeans that we may assume that there is a point-stabilizerNω in N satisfying

Nω = (s, . . . , s): s ∈ S

.

(We warn the reader that the parameter) in the current context corresponds to the paramm of Construction 3.2, and not to the parameter) of that construction.) We note that) is atleast two.

As M is minimal normal inG, M is characteristically simple. It is not elementaabelian as it is transitive of degree|S|)−1. ThusM ∼= T k for some non-abelian simplgroupT and positive integerk. Arguing as in the first paragraph of the proof of Lemmaof [31], we see thatM ∩ N = id, and asM is minimal normal inG we haveM N .Suppose first thatM contains one of the simple direct factors ofN , sayM containsS1.ThenS1 is a simple direct factor ofM , and sinceM is a minimal normal subgroup ofG,M is the product of theG-conjugates ofS1. Thus we may assume thatM = S1 × · · · × Sk ,


-

group

of

to

ith

ofg

with k ). SinceM is transitive of degree|S|)−1, it follows that eitherk = ) and (i) holds,or k = )− 1 and (ii) holds.

Thus we may assume thatM contains none of theSi . Let πi :N → Si denote theprojection map and letBi = πi(M), for 1 i l. Then by [28, Theorem 2 and Lemma 4.3],Bi < Si for eachi, and) 3. Moreover if) = 3 thenS or some automorphismgroup ofS has a strong triple factorisation where each of the factors has a normal subwhich is a power ofT . By [3, Theorem 1], there are so such factorisations. Hence) = 2. Inthis case by [28, Lemma 4.5], and arguing as in the proof of Lemma 6.2,M = B1 × B2 ∼=T × T and (T ,S) is (A5,A6) or (M11,M12) or (;7(q),P;+

8 (q)) (q 2). Now Mω =M ∩ Nω = (s, s): s ∈ B1 ∩ B2, and sinceM is transitive onΩ it follows thatB2 = Bσ

1and (4-A) holds, whereσ ∈ AutS, andS,T ,σ satisfy (i), (ii) or (iii) of Section 4.4. Sincethe centraliser ofB1 in AutS is trivial in each case, an easy computation shows thatM hastrivial centraliser inW(S,2,1). In particularM is the unique minimal normal subgroupG soG is quasiprimitive. It now follows that(G, H ) is a 4-inclusion. Corollary 6.7. TheSDcolumn of the Results Matrix(Fig. 1) holds.

Proof. By Lemma 6.6 applied with the quasiprimitive permutation groupG in place ofG, and withH in place ofH , we deduce that one of (i), (ii) and (iii) holds. We leave itthe reader to complete the proof by verifying that if (i) holds then(G,H) is a 1-inclusionwith X = SD; that if (ii) holds then(G,H) is a 3-inclusion belonging to subcase (1) wX = ASreg if ) = 2 and to subcase (3) withX = TW if ) > 2; and that if (iii) holds then(G,H) is a 4-inclusion withX = PA.

The next result depends on Theorem 4.7, and hence on the results of [4].

Lemma 6.8. TheSD andCD rows of the Results Matrix(Fig. 1), apart from theAS-column,hold.

Proof. Given Lemmas 6.1, 6.5 and 6.3 and Corollary 6.7, we may assume that(G,H)

is an inclusion withG quasiprimitive of type SD or CD and H primitive of type CDor PA. ThusH leaves invariant a non-trivial blow-up decompositionE of Ω in whichthe componentH1 is of type SD or AS, according to whetherH is of type CD or PArespectively. NowE is a non-trivial homogeneousG-invariant Cartesian decompositionG. Part (3) of Theorem 4.7 implies thatG is not of type SD, which verifies the remaininentries in the SD row. HenceG is of type CD. Part (4) of Theorem 4.7 implies thatE is ablow-up decomposition relative toG with componentG1 quasiprimitive of type SD or CD.It immediately follows that the inclusion(G,H) is a non-trivial blow-up of the inclusion(G1,H1), which serves to verify the remaining entries in the CD row. (We remark that theinclusion(G1,H1) is an (SD,SD)-inclusion if Y = CD, and an(SD,AS)-inclusion or a(CD,AS)-inclusion ifY = PA, since by Corollary 6.7 no(CD,SD)-inclusions exist.)

We now have enough information to handle the case whereY = CD.

Lemma 6.9. TheCD column of the Results Matrix(Fig. 1) holds.


l

e

f

phe

s

thf

s

-

n

Proof. We assume thatH is primitive of type CD: thusH leaves invariant a non-triviablow-up decompositionE of Ω of indexm 2 with componentsH1, . . . ,Hm which areprimitive subgroups of Sym(Γ ) of type SD. We may assume thatH Sym(Γ ) Sm, andwe note that

N = N1 × · · · × Nm Sym(Γ )m,

whereNi = SocHi∼= Sr for i = 1, . . . ,m, with ) = mr andr 2. By Lemma 6.2 we hav

M N and either case (1) or case (4) of that lemma holds.Suppose first that case (1) holds, that is,M contains a simple direct factorS of N . Then

S is a simple direct factor ofM, soS = T , M ∼= Sk , andM is a direct product of some othe simple direct factors ofN . SinceM is transitive onΩ , M must contain at leastr −1 ofthe simple direct factors ofNi for eachi. However, sinceM is a minimal normal subgrouof G, G permutes the simple direct factors ofM transitively by conjugation, and hence tnumber of simple direct factors ofNi contained inM is independent ofi. It follows thateitherM = N or M ∼= Sm(r−1). In the former case, a stabiliserMω

∼= Sm and it follows,sinceM is the unique minimal normal subgroup ofG, thatG has type CD; the correctnesof the(CD,CD)-cell of the Results Matrix follows from Lemma 6.8.

In the latter caseM is regular, non-abelian, and not simple, soG has type TW. ByLemma 3.7 we may assume thatG is permutationally isomorphic to a twisted wreaproductS twrφ P in its base group action, whereφ−1(InnS) is a core-free subgroup oP , andk = m(r − 1) is the index inP of the twisting subgroupQ, the domain ofφ. SinceM is a direct product of simple direct factors ofN , M is a normal subgroup ofN . We leaveit to the reader to verify that this only happens if(G,H) is a 3-inclusion (Subcase 4) arequired.

Suppose now that case (4) of Lemma 6.2 holds. Thenr = 2, and (T ,S) is as inLemma 6.2(3); in particular|T | < |S|. For i = 1, . . . ,m let pi be the projection mappi :N →Ni ; we have∣∣pi(M)

∣∣ |T |2 <∣∣pi(N)

∣∣ = |Ni | = |S|2 = |Γ |2.

As M is a transitive non-abelian non-regular minimal normal subgroup ofG we canapply Lemma 4.9(1) to the homogeneousG-invariant Cartesian decompositionE of Ω

and deduce that

M = p1(M)× · · · × pm(M).

Furthermore the initial part of Lemma 4.9(2) implies thatp1(M) is a transitive nonregular minimal normal subgroup of the componentG1 of G, and thatE is aG-transitivehomogeneousG-invariant Cartesian decomposition ofΩ . SinceG1 H1 and T = S,it follows from Lemma 6.6 that(G1,H1) is a 4-inclusion andp1(M) = SocG1. Nowthe permutation groupp1(M) = SocG1 on Γ is explicitly known, and direct calculatioshows thatCSym(Γ )(p1(M)) is trivial, whence Lemma 4.9(2) implies thatE is a blow-updecomposition forG, and so the inclusion(G,H) is a non-trivial blow-up of the inclusion(G1,H1) as required.


hatade

s

,

4.5ledge

s

s only

Lemma 6.10. ThePA column of the Results Matrix(Fig. 1) holds.

Proof. We assume thatH is primitive of type PA. By Lemma 6.8 we may assume tX is not SD or CD. Also by the explanatory note 6 following Fig. 1, no claims are mabout quasiprimitive-primitive inclusions of types (TW, PA) or (PA, PA). Thus we mayassume thatG has type AS, so M = SocG is a non-abelian simple groupT . Also H

leaves invariant a homogeneous Cartesian decomposition ofΩ of indexm 2, and wemay assume thatH is a subgroup ofH = Sym(Γ ) Sm acting onΩ = Γ m via the productaction of the wreath product. Note thatT G is certainly quasiprimitive, and so(T , H )

is a quasiprimitive-primitive inclusion. AsH is primitive of type PA, Lemma 6.2 impliethat

SocG = T SocH = (Alt (Γ )

)m (Sym(Γ )

)m.

Fix γ ∈ Γ and fori = 1, . . . ,m define the subgroupKi of T by

Ki = SocG∩ (s1, . . . , sm) ∈ (

Sym(Γ ))m: γ si = γ

.

The transitivity ofT onΩ = Γ m implies that

|Γ | = |T : Ka |, T = KaKb and T = Ka(Kb ∩Kc),

whenevera, b, c are distinct integers in1, . . . ,m. If m 3 then in the terminology of [3]K1,K2,K3 is a non-trivial ‘strong multiple-factorisation’ ofT with |K1| = |K2| = |K3|.Inspection of Table 5 of [3] shows that no such multiple-factorisations exist. Hencem = 2andK1, K2 are proper subgroups ofT satisfying

T = K1K2 and |K1| = |K2|.ThusT ,K1,K2 are known explicitly and the possibilities are as in (i)–(iv) of Section(see the discussion in the first part of that subsection). Observe also that knowof K1, K2 determinesT up to permutational isomorphism sinceK1 ∩ K2 is preciselythe point-stabilizer inT of the point (γ, γ ) ∈ Γ 2. Let σ1, σ2 be the projection map(Sym(Γ ))2 → Sym(Γ ) given by

σi : (s1, s2) → si for all s1, s2 ∈ Sym(Γ ).

On inspecting the definition of 5-inclusions we see that to prove the lemma, it remainto show that ifH is any primitive subgroup ofH = Sym(Γ ) S2 that containsT and is oftype PA, then

Soc(σ1

(H ∩ (

Sym(Γ ))2)) × Soc

(σ2

(H ∩ (

Sym(Γ ))2))

(6-A)

is a minimal normal subgroup ofH (and so equal to SocH ). By Proposition 4.5(iii) appliedto H , theH -invariant Cartesian decomposition ofΩ induced by the containment ofH inSym(Γ ) S2 is a blow-up decomposition, and (6-A) follows immediately.


f the

utnt. We

heing to

Noteny,f suchf typeroupsbcase

ave

st

tion.

s to

This finishes the proof of Theorem 1.2: it follows as an immediate corollary oresults 6.1, 6.3, 6.4, 6.5, 6.7, 6.9, and 6.10.

7. Final remarks

7.1. (AS,AS)-inclusions

Here we consider proper inclusions(G,H) whereG < H Sn with H primitive oftype AS andG quasiprimitive of type AS. The Results Matrix gives no information abothis case and the aim of this subsection is to identify the principal obstacles presestart by dividing into three disjoint subcases, namely,

(1) SocH = An;(2) SocG = SocH = An;(3) SocG = SocH = An.

Subcase1. The inclusions(G,Sn) and (G,AnG) always occur and are precisely tinclusions belonging to this subcase. We remark that the primitive inclusions belongthis subcase are described in this way in §3.1 of [31].

Subcase2. HereG andH have a common socle that is a non-abelian simple group.that for any primitive permutation groupH of type AS, its socle is transitive, and so agroupG satisfying SocH G<H is necessarily quasiprimitive of type AS. Furthermoreit is clear that all inclusions in this subcase arise in this way. Our understanding oinclusions is thus equivalent to our understanding of primitive permutation groups oAS, which in turn is equivalent to our understanding of the core-free maximal subgof almost simple groups. We remark that the primitive inclusions belonging to this suare described in this way in §3.7 of [31].

Subcase3. This is where the difficulties lie. The principal information that we hcan be summed up as follows. LetHω be a point-stabilizer inH ; asH is a primitivepermutation group of type AS, the subgroupHω is a maximal subgroup of the almosimple groupH and Hω does not containN := SocH . SinceG is a quasiprimitivepermutation group of type AS, it follows from Lemma 6.2 thatM := SocG < N .Consequently, we have the factorisations

H = GHω, H = MHω and N = MNω

of the almost simple groupH and its socleN .We would like to make use of the results of [26] to yield information about this situa

If G were maximal inH subject to not containingN , thenH = GHω would be listed in[26] or [27]. Thus such inclusions(G,H) are essentially classified. The problem then ideal with the case in whichG is not maximal in this sense. Suppose then that

G<L<H


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with L maximal inH subject to not containingN . ThenH = LHω is listed in [26] or[27], so we may regard the inclusions(L,H) as known, and we have a further propfactorisationL = GLω. Suppose thatL is quasiprimitive onΩ . ThenL is almost simpleby [31, Proposition 6.2] ifL is primitive (since SocH = An and sinceL contains aquasiprimitive group of type AS), or by Theorem 1.2 applied to the quasiprimitivprimitive inclusion(L,H) if L is quasiprimitive and imprimitive. We may apply similreasoning to the factorisationL = GLω of the smaller almost simple groupL noting thatLω may not be maximal inL. On the other hand ifL is not quasiprimitive onΩ , thenL has a non-trivial intransitive normal subgroup. LetK be a maximal intransitive normasubgroup ofL. As G is quasiprimitive we haveG ∩ K = id whenceL/K has a propefactorisation(GK/K)(LωK/K) with GK/K ∼= G. Analysing this smaller factorisatiocarefully leads to a complete classification of all possible(AS,AS)-inclusions. This workis being undertaken by Liebeck, Saxl and the second author and will be publishedcourse.

7.2. Other entries

The only entries of the Results Matrix that fail to give any information concerningrelevant inclusions, apart from the (AS, AS)-entry, are the (TW, PA)- and (PA, PA)-entries.As mentioned previously, the results of [4] yield a great deal of information aboucorresponding inclusions.

However [4] does not utilize the following observation about (TW, PA)-inclusions. Inorder to state the observation we suppose that(G,H) is a (TW, PA)-inclusion, and recalthatM = SocG andN = SocH are both non-abelian and characteristically simple,M N and thatM is regular. It follows thatN = NωM is an exact factorisation ofNwhereNω is any point-stabilizer inN , that isNω ∩ M = id. Thus, if a theory of exacfactorisations of non-abelian characteristically simple groups could be developedfurther progress could be made in classifying (TW, PA)-inclusions.

7.3. Case distinctions

Throughout this paper we have considered eight types of primitive permutation gnamely types HA, . . . ,PA and eight types of quasiprimitive permutation groups, namtypes HA, . . . ,PA. Our reason for choosing these eight types of primitive permutagroups is very much as stated in [31], that such a division has proved the mostin practice. Indeed, the results of [31] and the O’Nan–Scott Theorem as stated ishow that these types are naturally distinguished, either by the structure and the acthe socle, or by the nature of the primitive overgroups. On the other hand, our reasochoosing the eight types of quasiprimitive permutation groups are not well-groundexperience, and rely more upon analogy. It is entirely possible that further subdivisour given classes of quasiprimitive permutation groups may be appropriate. An imaim of this paper is to augment our understanding of quasiprimitive permutation gby eliciting further similarities and differences between the current classes with a via possible sharpening of the quasiprimitive version of the O’Nan–Scott Theorem inFor example, we have already seen fit to divide the quasiprimitive permutation gro


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type AS into two, namely types ASreg and AS¬reg. This can be justified in that the grouof type ASreg share many characteristics in common with those of type TW, as can be seefrom the fact that 3-inclusions apply only to types ASreg and TW. Other distinctions thamay prove sensible are primitive versus imprimitive, and the existence, or otherwiblow-up decompositions or other invariant Cartesian decompositions.

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on primitive overgroups of quasiprimitive permutation groups

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