bases of primitive linear groups

Journal of Algebra 252 (2002) 95–113

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Bases of primitive linear groups

Martin W. Liebecka,∗ and Aner Shalevb

a Department of Mathematics, Imperial College, London SW7 2BZ, England, UKb Mathematics Institute, Hebrew University, Jerusalem 91904, Israel

Received 25 September 2001

Communicated by Jan Saxl

Abstract

Let V be a finite vector space andG GL(V ) a linear group. A base ofG is a set ofvectors whose pointwise stabiliser inG is trivial. We prove that ifG is irreducible andprimitive onV , thenG has a base of size at most 18 log|G|/log |V | + c, wherec is anabsolute constant. This verifies part of a conjecture of Pyber on base sizes of primitivepermutation groups. 2002 Elsevier Science (USA). All rights reserved.

1. Introduction

LetG be a permutation group on a finite setΩ of sizen. A subset ofΩ is saidto be abasefor G if its pointwise stabiliser inG is trivial. The minimal size of abase forG is denoted byb(G). A well known conjecture of Pyber [12] states thatthere is an absolute constantc, such that ifG is primitive onΩ , then

b(G) < clog|G|logn

.

Note that |G| nb(G), so certainlyb(G) log|G|/logn. Pyber’s conjecturestrengthens a conjecture of Babai, which was proved in [4].

The authors acknowledge the support of the Asymptotic Group Theory Programme of theInstitute of Advanced Study, February–August 2000.

* Corresponding author.E-mail address:[email protected] (M.W. Liebeck).

0021-8693/02/$ – see front matter 2002 Elsevier Science (USA). All rights reserved.PII: S0021-8693(02)00001-7

96 M.W. Liebeck, A. Shalev / Journal of Algebra 252 (2002) 95–113

Making inroads into Pyber’s conjecture has proved fairly tough. A proof of it inthe case whereG is almost simple can be found in [9, Theorem 1.3], supplementedby [2]; and a reduction theorem to the cases whereG is affine or almost simple isforthcoming in [13]. Thus the focus is now on primitive affine permutation groups;heren= pm for some primep andG= VH , whereV ∼= (Cp)m is an elementaryabelian regular normal subgroup ofG, andH = G0 is a point-stabiliser and anirreducible subgroup ofGL(V ) ∼= GLm(p). Writing b(H) for the minimal sizeof a base for the action ofH on the set of vectors inV , we haveb(G)= 1+b(H),so the problem becomes one of finding a base of suitable size for an irreduciblelinear group.

Various special cases of the problem have appeared: the case whereH (hencealsoG) is solvable has been handled in [14], and that in whichH is ap′-groupin [3] (in both casesH has a base of bounded size).

In this paper we prove a strong form of Pyber’s conjecture (with an explicitconstant) in the case in whichH acts primitively as a linear group on thevector spaceV (in other words,H does not preserve any non-trivial direct sumdecomposition ofV ). Our main result is

Theorem 1. There is an absolute constantC such that ifH GL(V ) is an irre-ducible, primitive linear group on a finite vector spaceV , then either

(i) b(H) C, or(ii) b(H)< 18 log|H |/ log|V | + 27.

The constantC in Theorem 1 is explicitly defined below (after the statementof Theorem 2).

Theorem 1 follows from a more detailed result, Theorem 2, which describes thestructure of irreducible primitive linear groupsH for which b(H) is unbounded.This result may prove useful in an inductive approach to the conjecture for theremaining case of imprimitive linear groups.

For the statement of Theorem 2 we require some notation. For a prime powerpa and a positive integerd , let Vd(pa) denote a vector space of dimensiondover the fieldFpa of orderpa . Denote byCln(pe) a quasisimple classical groupwith natural moduleVn(pe), i.e., one of the groupsSLn(pe), SUn(pe/2), Spn(p

e),Ωn(p

e). If Fpf is an extension field ofFpe then byCln(pe,pf ) we mean asubgroup ofΓ Ln(pf ) whose generalised Fitting subgroup isCln(pe), actingnaturally onVn(pf ). Writing X = Cln(pe), we have

X Cln(pe,pf

)NΓLn(pf )(X)=

(F∗pfNGLn(pe)(X)

).f.

Observe also thatCln(pe,pf ) embeds naturally inGLnf (p) via

Cln(pe,pf

) Γ Ln

(pf

)GLnf (p).

M.W. Liebeck, A. Shalev / Journal of Algebra 252 (2002) 95–113 97

Next, by the natural module overFpf for the alternating group Altm we meanthe non-trivial irreducible constituent of the usualm-dimensional permutationmodule overFpf ; it has dimensionm− δ(p,m), whereδ(p,m) is 2 if p |m andis 1 otherwise. By(Altm,pf ), we mean a subgroup ofΓ Lm−δ(p,m)(pf ) whichhas generalised Fitting subgroup Altm in this representation; thus

Altm (Altm,p

f)

(F∗pf

Symm).f < Γ Lm−δ(p,m)

(pf

) GL(m−δ(p,m))f (p).

If V,W are vector spaces over the same field, andX GL(V ),Y GL(W),then byX ⊗ Y we mean the image ofX × Y acting in the natural way on thetensor productV ⊗W .

ForH GL(V ), defineb∗(H) to be the minimal size of a setB of vectorssuch that any element ofH which fixes every 1-space〈v〉 with v ∈B is necessar-ily a scalar multiple of the identity. We call such a setB a strong baseof H . Anelementary argument (see Lemma 3.1) shows that

b(H) b∗(H) b(H)+ 1.

Finally, recall that for a finite groupH , the generalised Fitting subgroup ofHis denoted byF ∗(H). We haveF ∗(H)= F(H)E(H), whereF(H) is the Fittingsubgroup andE(H) is the subgroup generated by all quasisimple subnormalsubgroups ofH .

Theorem 2. LetV = Vd(p) be a vector space of dimensiond overFp (p prime),and letH GL(V ). Suppose the representation ofH on V is irreducible andprimitive. LetC be the constant in the statement of Theorem1.

(i) If b∗(H) > C, then

H H0 ⊗s⊗1

(Altmi ,p

ai) ⊗

t⊗1

Cldi(pei ,pfi

),

wheres + t 1 and(1) H0 GLd0(p) with b∗(H0) C,(2) each (Altmi ,p

ai ) is embedded inGL(mi−δ(p,mi))ai (p) as above, andm1< · · ·<ms ,

(3) each Cldi (pei ,pfi ) is embedded inGLdifi (p) as above, andd1 < · · ·

< dt ; in particular, d = d0∏s

1(mi − δ(p,mi))ai∏t

1difi ,(4) F ∗(H) contains

⊗s1 Altmi ⊗

⊗t1 Cldi (p

ei ).

(ii) LetH be as in(i), with b∗(H) > C. If ms dt thend < (dtft )2, and

b∗(H) b∗(GLd/dtft (p)⊗ Cldt(pet ,pft

)) 9

etd2t

d+ 21,


and ifms > dt then, writingm′s =ms − δ(p,ms), we haved < (m′

sas)2 and

b∗(H) b∗(GLd/m′sas(p)⊗ (

Altms ,pas

))

3ms logp msd

+ 18.

The deduction of Theorem 1 from Theorem 2 is given at the end of the paper.In the case whereE(H) is quasisimple, we actually prove a stronger result

than Theorem 2 in Proposition 2.2 below (the latter result does not require thehypothesis of primitivity ofH onV ).

The constantC in Theorems 1 and 2 is defined as follows. By Lemma 3.6below, ifH is as in the hypothesis of Theorem 1, and the Fitting subgroupF(H)

is irreducible onV , thenb(H) is bounded; define

C1 = maxb∗(H): H GL(V ) primitive,F(H) irreducible

.

Next, givend,p, with p prime, writeCd,p for the collection of subgroupsHof GLd(p) for which E(H) is quasisimple and irreducible as a subgroup ofGLd/f (p

f ) for somef dividing d . By Proposition 2.2 below, ifH is such asubgroup thenb(H) is bounded unlessH = Cld/f (pe,pf ) or (Altm,pf ), wheree | f or d/f =m− δ(p,m), respectively. WriteC ′

d,p for the set of subgroups inCd,p which are not of the latter types. Now define

C2 = maxb∗(H): H GLd(p) for somed,p, andH ∈ C ′

d,p

.

The constantC is defined by

C = maxC1,C2,33.Almost all of the rest of the paper consists of the proof of Theorem 2, and

is divided into three further sections. In Section 2 we handle the case where thelinear groupG is quasisimple, which, not surprisingly, is a major part of the proofof Theorem 2. It turns out that much more than just the bound onb(H) in theconclusion of Theorem 2 is true in this case; indeed, in Proposition 2.1 we showthat ifH is a quasisimple group of Lie type, andV is anH -module of sufficientlylarge dimension (for example, of dimension at leastm2 if H is a classical groupClm(q) of large dimensionm), thenH actually has a regular orbit onV (orequivalently,b(H)= 1). The bound of Theorem 2 forH quasisimple in general,is deduced from this in Proposition 2.2. Section 3 contains a number of lemmasmainly concerning base sizes of tensor product actions. Finally in Section 4, wecomplete the proof of Theorem 2 and deduce Theorem 1.

2. Bases for quasisimple linear groups

In [6, Theorem 6], it is shown that ifH is a quasisimple group andV is anon-trivial irreducibleH -module over a field of characteristicp, then one of thefollowing holds:


(1) H has a regular orbit on the vectors ofV ;(2) H =Ac andV is the natural module forH ;(3) H is a group of Lie type in characteristicp;(4) (H,V ) is one of finitely many exceptions.

In case (2) there are characteristicsp in whichH has no regular orbits. Thus forthe purposes of studying regular orbits, attention focuses on case (3). The nextresult shows that for representations of sufficiently large dimension of groups incase (3), there are regular orbits.

By theuntwisted Lie rankof a finite group of Lie type, we mean the Lie rankof the corresponding simple algebraic group.

Proposition 2.1. LetH be a quasisimple group of Lie type in characteristicp,and letV be an absolutely irreducibleH -module over a field of characteristicp.

(i) There is a functionf :N → N such that ifH has untwisted Lie rankl anddimV > f (l), thenH has a regular orbit onV .

(ii) Suppose thatH is a classical group with natural module of dimensionm, andthatdimV >m2. Then providedm is sufficiently large,H has a regular orbitonV .

Note that part (ii) gives an explicit form for the functionf in part (i) for groupsof large rank.

Proof. (i) This part is essentially already established in the proof of [6, Theo-rem 6]. LetH =Hl(q0), a group of rankl overFq0, and letF be the field overwhich theH -moduleV is defined. Assume thatH has no regular orbit onV .As in [6, Step 1, p. 454],F is finite, sayF = Fq , and we have|V |< |H |8(2l+1).

Inspection of the order formulae for groups of Lie type gives|H |< q4l20 . Setting

s = logq0/ logq , these inequalities imply that

dimV < 32l2(2l + 1)s.

If s 4 then we have the conclusion of (i). Now supposes > 4. Then [7, 5.4.6and 5.4.13] imply that dimV ls , and hence

qls |V |< q32l2(2l+1)s.

As s > 4, the inequalityls < 32l2(2l + 1)s shows thatl ands are bounded. Thiscompletes the proof of (i).

(ii) Write H = Clm(q0), a classical group overFq0 with natural module ofdimensionm, and letF be the field over whichV is defined. The conclusion istrivial if F is infinite, so assume thatF is finite, sayF = Fq .

If H has no regular orbit thenV = ⋃h∈H( CV (h). So in order to show thatH

has a regular orbit, it is sufficient to prove that


|V |>∑h∈H(

∣∣CV (h)∣∣. (∗)

Forh ∈H −Z(H), let α(h) be the smallest number of conjugates ofh whichgenerateH . Writing [V,h] = 〈v − vh: v ∈ V 〉, we have

α(h) dimV

dim[V,h] .

Since also dim[V,h] = dimV − dimCV (h), it follows that

dimCV (h)(

1− 1

α(h)

)dimV. (1)

The key result we shall use is taken from [10, 1.10], which states that there is anabsolute constantd such that forh ∈H −Z(H),

α(h) < dlog|H |log|hH | . (2)

For a constantc, define

Ec = h ∈H :

∣∣hH ∣∣< qcm0

.

Since by [8, Theorem 1], the number of conjugacy classes inH is less thanq3m0 ,

we have|Ec| < q(3+c)m0 . From the order formulae for simple groups we have

|H | < q4m2

0 . Hence by (2), forh ∈ H − Ec we haveα(h) < 4md/c, while forh ∈Ec −Z(H) we haveα(h) < 16m by [6, Theorem 2]. By (1), it follows that

dimCV (h) <(1− c

4md

)dimV, if h ∈H −Ec,

dimCV (h) <

(1− 1

16m

)dimV, if h ∈Ec −Z(H).

Write n= dimV . If nm3/9, then [11, 5.1] implies thatnm2 providedmis sufficiently large, contrary to hypothesis. Hence we may assume thatn >m3/9.

Let s = logq0/ logq . Then [7, 5.4.6] implies thatnms .We now bound the sum in (∗). Write this sum asΣ1 +Σ2, where

Σ1 =∑

h∈Ec−Z(H)

∣∣CV (h)∣∣, Σ2 =∑

h∈H−Ec

∣∣CV (h)∣∣.From the above we have

Σ1< |Ec|qn(1−(1/16m)) < |V |q(3+c)ms · min(q−m2/144, q−ms−1/16).

Also, if s 3 then

Σ2< |H |qn(1−(c/4dm)) < q4m2

0 |V |q−nc/4dm |V |q12m2−m2(c/36d),


while if s > 3 then

Σ2< q4m2

0 |V |q−nc/4dm < |V |q4sm2−ms−1(c/4d).

In all cases, we see that provided we choosec to be greater than 432d (= 12·36d),we haveΣ1 +Σ2< |V |. This establishes (∗) and completes the proof.

The next proposition establishes Theorem 2(i) in the case whereE(H) is qua-sisimple and irreducible (whereE(H) is the group generated by all quasisimplesubnormal subgroups ofH ); indeed, the proposition is stronger than Theorem 2for this case, since the hypothesis does not requireH to be primitive onV .

Proposition 2.2. Let V = Vd(q) (q = pe) andH GL(V ), and suppose thatE(H) is quasisimple and absolutely irreducible onV . Then one of the followingholds:

(i) b(H) is bounded;(ii) E(H) = Altm and V is the naturalAltm-module overFq , of dimension

d =m− δ(p,m);(iii) E(H)= Cld (q0), a classical group with natural module of dimensiond over

a subfieldFq0 of Fq .

Proof. Suppose thatb(H) is unbounded. Then certainlyd = dimV is un-bounded.

We first show thatb(E(H)) is also unbounded. Suppose to the contrary thatE(H) has a baseB onV of bounded size, and letH0 be the pointwise stabiliserof B in H . ThenH0 ∼= H0E(H)/E(H). Since alsoH0 ∩ Z(H) = 1, it followsthatH0 is isomorphic to a subgroup of Out(E(H)). From the structure of outerautomorphism groups of simple groups (see, for example, [16, Theorems 30, 36]),it follows thatH0 has a series 1=N0 N1 · · · Nk =H0 with k 5, such thateach factor groupNi+1/Ni cyclic. By [15, 3.1], a cyclic subgroup ofGL(V )necessarily has a base of size 1 onV . Hence we easily see thatH0 has a base ofsize at mostk 5, whenceb(H) is bounded, a contradiction.

Thereforeb(E(H)) is unbounded. In particular,E(H) has no regular orbitson V . It follows from [6, Theorem 6] that either (ii) holds, orE(H) is of Lietype in characteristicp. Assume the latter. Then by Proposition 2.1, we haveE(H)= Clm(q0) with m large, and dimV m2.

Now [11, 5.1] determines all irreducibleFpE(H)-modules of dimension lessthanm3/8. From this we see thatV = V ⊗Fp is one the following modulesV (λ),whereV (λ) denotes the irreducibleFpE(H)-module of high weightλ, whereλis given up to automorphisms ofE(H) (we use standard notation for weights, asin [7, §5.4], for example; andLεm(q0) denotesPSLm(q0) if ε = 1 andPSUm(q0)

if ε = −1):


E(H)/Z(E(H)) λ

any λ1any λ2any(p = 2) 2λ1any λ1 + piλ1 (i > 0)Lεm(q0) λ1 + λm−1Lεm(q0) λ1 + piλm−1

(Note that it is sufficient to consider the possibilities forλ up to automorphismsof E(H), since the image group inGL(V ) is unaffected by application of suchan automorphism.)

Application of [7, 5.4.6] shows that in all but two cases,Fq0 is a subfieldof Fq . The exceptional cases occur forλ= λ1 + qλ1, and forλ= λ1 +λm−1 withE(H)/Z(E(H))= Um(q0); in both casesFq0 is a subfield ofFq2. We postponethese exceptional cases until the end of the proof; so until then, suppose thatFq0

is a subfield ofFq . LetW0 = Vm(q0) be the natural module forE(H), and setW =W0 ⊗ Fq = Vm(q).

If λ = λ1 then V = W , the natural module, and conclusion (iii) of theproposition holds.

We shall show that for all the other possibilities forλ, E(H) has a boundedbase onV .

First supposeλ= λ2. Here dimV (λ)= (m2

)− αH , whereαH = 0 unless eitherE(H) is symplectic or it is orthogonal withp = 2; in these casesαH = 2 if p di-videsm/2, andαH = 1 otherwise (all this is well known—see [11], for example).

Consider first the cases whereαH = 0. HereV = ∧2W , andE(H) lies in theaction ofSL(W) on this space. It is convenient to replaceV with the dual space∧2W∗ (which we can do by applying a suitable automorphism ofE(H) to theaction onV ). This space∧2W∗ can be identified with the space of alternatingbilinear forms onW . Hence ifm = dimW is even, there is a vectorv ∈ V suchthatSL(W)v = Sp(W); indeed, if we writem= 2k and letw1, . . . ,w2k be a basisforW , thenv = ∑k

1wi ∧wk+i fits the bill. Then the action ofSL(W) on the orbitof v is its action on the cosets ofSp(W), which by [9, 1.3] has a bounded base.If m= 2k+ 1 is odd, take a basisw1, . . . ,w2k, d and definev as above, and alsov1 =w1∧d, v2 =w2∧d . It is easy to check thatSL(W)v,v1,v2 fixes〈w1, . . . ,w2k〉and〈d〉, inducingSp2k(q) on the former, and now the above argument shows thatthere is a bounded base.

Now supposeE(H) is a symplectic group. HereE(H) lies in the action ofSp(W) onV , which is a section of∧2W as we shall now exhibit. Writem= 2k,let ( , ) be a non-degenerate alternating bilinear form onW fixed bySp(W), and letei, fi (1 i k) be a standard basis forW . One checks that there is anSp(W)-invariant symmetric bilinear form( , )1 on∧2W defined by

(v ∧w,v′ ∧w′)1 = (v, v′)(w,w′)− (v,w′)(w,v′)


for v,w,v′,w′ ∈W . Moreover,Sp(W) fixes the vectorf = ∑k1 ei ∧ fi . Observe

that(f,f )1 = k; sof is singular if and only ifp | k. If p k thenV = f⊥, whileif p | k thenV = f⊥/〈f 〉.

Let r = [12k

], and chooseβ ∈ Fq such that

v =r∑1

ei ∧ fi + βer+1 ∧ fr+1 ∈ f⊥.

A simple calculation (see the proof of Lemma 3.3(i) below) shows thatSp(W)vfixes the subspaceX, where

X= 〈ei , fi : 1 i r + 1〉 if β = 0,

〈ei , fi : 1 i r〉 if β = 0.

HenceSp(W)v lies in a subgroupM = Sp2(r+1)(q)×Sp2(k−r−1)(q) or (Sp2r (q)×Sp2(k−r)(q)).δ, whereδ = 2 if k is even andδ = 1 if k is odd. The argument inthe first part of the proof of [9, 4.5] shows that ifx is an element of prime order inSp(W), then|xSp(W)∩M|< |xSp(W)|3/4+ε , whereε is an arbitrarily small positiveconstant. Using this, the proof of [9, Theorem 1.3, p. 501] shows thatSp(W) has abase of bounded size in its action on the cosets ofM. HenceE(H) has a boundedbase onV , as required.

Finally for this case (λ = λ2), if E(H) is an orthogonal groupΩ(W) incharacteristic 2, thenΩ(W) lies in Sp(W) in its action onV (λ2), so the previousparagraph shows that there is a bounded base.

Now suppose thatλ = 2λ1 with p = 2. If E(H) is not an orthogonal groupthenV = S2W . As before we replaceV by the dual spaceS2W∗, which can beidentified with the space of symmetric bilinear forms onW , from which we seethat there is a vectorv ∈ V such thatSL(W)v = O(W), an orthogonal group.Hence we deduce from [9, 1.3] again thatE(H) has a bounded base onV . Andif E(H) is orthogonal, then dimV (2λ1) is 1

2m(m+ 1)− βH , whereβH = 2 ifp | m andβH = 1 otherwise, and an argument very similar to that given for thesymplectic groups in the previous case yields a bounded base.

Next consider the case whereE(H)/Z(E(H))= Lm(q0) andλ= λ1 + λm−1.HereV is the space ofm×m matrices overFq of trace zero (modulo the scalarmatrices ifp | m), and the action ofSL(W) ∼= SLm(q) is by conjugation (andcontainsE(H)). Let r = [1

2m]

and define anm×m matrix

v = diag(Ir ,−r,0, . . . ,0).ThenGL(W)v is contained in a subgroup(GLr(q)×GLm−r (q)).δ (δ = 1 or 2)if −r = 1 in Fq , and inGLr+1(q)×GLm−r−1(q) if −r = 1. Now we see just asin the symplectic case forλ= λ2 thatSL(W) has a bounded base.

The case whereλ = λ1 + piλ1 or λ1 + piλm−1 is similar: here we canidentify V with the space of allm × m matrices overFq , the action ofg ∈SL(W) ∼= SLm(q) being to sendA→ gTAg(p

i) or g−1Ag(pi) for A ∈ V . Taking


A= diag(Ir ,0, . . . ,0) wherer = [12m

], we see as above that there is a bounded

base.It remains to deal with the two cases postponed at the beginning of the proof:

λ = λ1 + qλ1 or λ = λ1 + λm−1, with E(H)/Z(E(H)) = Um(q0) in the lattercase. In the first caseE(H) lies inSLm(q2) acting onW ⊗W(q) realised overFq ,whereW = Vm(q2). Let x1, . . . , xm be a basis forW . Fix α ∈ Fq2 with α /∈ Fq .

Then anFq -basis ofV is vi,wij ,w′ij (1 i, j m, i < j), where

vi = xi ⊗ xi, wij = xi ⊗ xj + xj ⊗ xi,w′ij = αxi ⊗ xj + αqxj ⊗ xi.

Definev = ∑m1 vi . Calculating as in the proof of Lemma 3.3(iii) below, we see

that

SLm(q2)

v= A⊗A−T: A ∈ SLm

(q2),

whence it is easy to see thatSLm(q2)v = SUm(q). Now [9, 1.3] shows that thereis a bounded base.

Finally, in theλ = λ1 + λm−1 case,E(H) lies in SUm(q) = g ∈ SLm(q2):g(q)gT = I acting by conjugation on theFq -space

V = A ∈Mm

(q2): tr(A)= 0, At =A(q)/scalars.

TakingA= diag(Ir ,−r,0, . . . ,0) wherer = [12m

], we have

SUm(q)A GUr(q)×GUm−r (q) or GUr+1(q)×GUm−r−1(q),

and we see in the usual way using [9] that there is a bounded base. This completesthe proof of the proposition.

3. Tensor product lemmas

In this section we prove a number of lemmas about bases of linear groups,mainly concerning linear groups in tensor product actions. Throughout,V willdenote a finite-dimensional vector space over a finite field.

Lemma 3.1. If H GL(V )=GLd(q) thenb(H) b∗(H) b(H)+ 1.

Proof. Since a strong base forH is also a base, we trivially haveb(H) b∗(H).Suppose now thatB = v1, . . . , vb is a base forH , and letH0 be the subgroupof H which fixes each of the 1-spaces〈vi〉 (1 i b). If h ∈ H0 then for1 i b there existsλi ∈ F∗

q such thath(vi) = λivi . SinceB is a base, the

maph→ (λ1, . . . , λb) fromH0 to (F∗q)b is a monomorphism. HenceH0 consists

of semisimple elements. By [15, 3.1] therefore,H0 has a regular orbit on the setof 1-spaces inV , and henceb∗(H) b+ 1. This completes the proof.


Lemma 3.2. If H Γ Ld(q), thenb(H) b(H ∩GLd(q))+ 1.

Proof. Let V = Vd(q), and letv1, . . . , vb ∈ V be a base forK :=H ∩GLd(q).ThenHv1,...,vb embeds inH/K, hence is cyclic; leth be a generator. By [5, 7.2],h is GLd(q)-conjugate to a field automorphism, fixing a basis, sayx1, . . . , xd ,of V . If λ is a generator ofF∗

q , then〈h〉λx1 = 1, hencev1, . . . , vb, λx1 is a basefor H .

Next we prove a key lemma about base sizes of tensor product groups.

Lemma 3.3. LetV1,V2 be vector spaces overFq with dimVi = ni andn1 n2,and letHi GL(Vi) for i = 1,2. Denote byH1 ⊗ H2 the image ofH1 × H2acting in the natural way on the tensor productV1 ⊗V2. Then the following hold.

(i) Let x1, . . . , xa ∈ V1 and y1, . . . , ya ∈ V2 be linearly independent vectors,and defineW1 = 〈x1, . . . , xa〉 andW2 = 〈y1, . . . , ya〉. If v = ∑a

1 xi ⊗ yi ∈V1 ⊗ V2, then(

GL(V1)⊗GL(V2))vGL(V1)W1 ⊗GL(V2)W2.

(ii) We have

b∗(H1 ⊗H2) max(b∗(H1), b

∗(H2)).

(iii) If n1 b∗(H2), then

b(H1 ⊗H2) 3

(1+ b∗(H2)

n1

).

In particular, b(GL(V1)⊗GL(V2)) 3(1+ (n2 + 1)/n1).(iv) If n1 b∗(H2), thenb(H1 ⊗H2) 3.

Proof. (i) Extend thexi andyi to basesx1, . . . , xn1 of V1 andy1, . . . , yn2 of V2.Let g⊗ h ∈ (GL(V1)⊗GL(V2))v , and for 1 i a let

xig =n1∑j=1

αij xj , yih=n2∑j=1

βij yj .

Then

v = v(g⊗ h)=a∑i=1

n1∑j=1

n2∑k=1

αij βik(xj ⊗ yk).

Therefore for 1 j a we havea∑i=1

αij βij = 1,


while for eitherj = k, or for j = k > a, we have

a∑i=1

αij βik = 0.

Definea× a matricesA= (αij ), B = (βij ). ThenATB = I . Fork > a, letBk bethea×a matrix obtained fromB by replacing the last column by(β1k, . . . , βak)

T.ThenATBk = diag(1, . . . ,1,0). AsA is invertible, it follows that

Bk = (AT )−1 diag(1, . . . ,1,0),

which is a matrix having last column(0, . . . ,0)T. Henceβik = 0 for 1 i a.Similarlyαik = 0 for 1 i a. In other words,g fixesW1 andh fixesW2, givingthe conclusion.

(ii) This is proved in [9, p. 504].(iii) Supposen1 b∗(H2), and lety1, . . . , yb be a linearly independent strong

base forH2 in V2, whereb= b∗(H2). Let x1, . . . , xn1 be a basis ofV1.Write b= rn1 + s with r, s integers and 0 s < n1. For 1 i r define

vi =n1∑k=1

xk ⊗ y(i−1)n1+k and vr+1 =s∑k=1

xk ⊗ yrn1+k.

Consider the stabiliserK = (H1 ⊗ H2)v1. By (i), this lies inH1 ⊗ (H2)W ,whereW = 〈y1, . . . , yn1〉. Calculating with matrices relative to the basisxi ⊗ yjof V1 ⊗W (1 i, j n1), we see that

KV1⊗W = A⊗A−T: A ∈GLn1(q)

.

It is well known thatPSLn1(q) is 2-generated and the Frattini subgroup ofSLn1(q)

coincides with its center. HenceSLn1(q) is also 2-generated and we may choosetwo matricesC,D ∈GLn1(q) which generateSLn1(q). Now definex = C ⊗ 1,y =D⊗ 1 ∈GL(V1 ⊗W). ThenK ∩Kx ∩Ky acts trivially onV1 ⊗W ; in otherwords, there are three vectorsv1, v1x, v1y ∈ V1 ⊗W such that the stabiliser inH1 ⊗H2 of these vectors acts trivially onV1 ⊗W .

Repeating this argument with the vectorsv2, . . . , vr+1, we see that there are3(r + 1) vectors inV1 ⊗ V2 such that the stabiliser of these vectors inH1 ⊗H2fixes each of vectorsxi ⊗ yj (1 i n1, 1 j b). It follows that this stabiliseracts as a scalar onV2, and hence acts trivially on the whole ofV1 ⊗V2. ThereforeH1 ⊗H2 has a base of size at most 3(r + 1), completing the proof of (iii).

(iv) Supposen1 b∗(H2). Let b = b∗(H2), and lety1, . . . , yb be a strongbase forH2. Sinceb n1 n2, we can extend this to a linearly independentsety1, . . . , yn1 in V2. Now define

v =n1∑i=1

xi ⊗ yi.


Arguing as in the proof of (iii), we see that the stabiliser inH1 ⊗H2 of v and twofurther vectors fixes eachxi⊗yi (1 i n1), and hence acts trivially onV1⊗V2.Thereforeb(H1 ⊗ V2) 3. Corollary 3.4. LetV = Vd(q) and letFq0 be a subfield ofFq with q = qf0 . If His a subgroupGLd(q0) ofGL(V ), thenb(H) 3(1+ d/f ).

Proof. RegardingV asVdf (q0), we have

H =GLd(q0)⊗ 1 GLd(q0)⊗GLf (q0)GLdf (q0).

Hence the conclusion follows from parts (iii), (iv) of the previous lemma.Lemma 3.5. LetV = V1 ⊗ · · · ⊗ Vt , where eachdimVi =m 2 and t 2, andlet G = (

⊗t1GL(Vi)).St , a subgroup ofGL(V ) (in the Aschbacher classC7,

see[7]). Thenb(G) 4.

Proof. Identify eachVi with V1, and letx1, . . . , xm be a basis ofV1. Definev = ∑m

1 xi ⊗ · · · ⊗ xi ∈ V , and writeB for the “base-group”⊗t

1GL(Vi). Asin the proof of Lemma 3.3(iii), we find that ift = 2 then

Gv ∩B = A⊗A−T: A ∈GLm(q)

,

while if t 3 then

Gv ∩B A⊗A⊗ · · · ⊗A: A ∈GLm(q), AAT = I.

Let SLm(q)= 〈C1,C2〉 and writeci = Ci ⊗ 1⊗ · · · ⊗ 1 for i = 1,2. If v1 = vc1,v2 = vc2, then it is easy to see thatGv,v1,v2 = St−1, permuting the lastt − 1vectors in each tensor in the natural way. Now let

y = x1 ⊗ · · · ⊗ x1 ⊗ x2 + x1 ⊗ · · · ⊗ x1 ⊗ x2 ⊗ x2 + · · ·+ x1 ⊗ x2 ⊗ · · · ⊗ x2.

ThenGv,v1,v2,y = 1, which completes the proof.Lemma 3.6. SupposeH is a primitive subgroup ofGLd(q) such that the FittingsubgroupF(H) is irreducible onV . Thenb(H) is bounded.

Proof. Write F = F(H), and letV = Vd(q) be the underlying vector space.RegardingV as a vector space over the field extension EndF (V ), we may takeit that F is absolutely irreducible onV and H NΓL(V )(F ). The result istrivial if F is abelian, so assumeF is non-abelian. WriteF = F1 × · · · × Fr ,a direct product of Sylow subgroups. ThenV = V1 ⊗ · · · ⊗ Vr with eachVian absolutely irreducibleFi -module, andNGL(V )(F )= ⊗r

1NGL(Vi)(Fi) (see [7,4.4.3]). Therefore by Lemma 3.3(ii), we may assume thatr = 1, in other words


F is ans-group for some primes. SinceH is primitive, it follows from Clifford’stheorem that every characteristic abelian subgroup ofF acts homogeneouslyonV , hence consists of scalars and is cyclic. ThusF is ans-group of symplectictype (see [1, p. 109]), the structure of which is given by a theorem of Philip Hall[1, 23.9]. It follows from this that we may takeF to be a central product ofan extraspecial groups1+2m (of exponents or 4) and a group of scalars, withdimV = sm.

If ρ :F → GL(V ) is the original representation, andα ∈ CAut(F )(Z(F )),then by the representation theory of symplectic type groups,αρ is equivalentto ρ, and hence there is an elementg ∈ NGL(V )(F ) inducingα on F . WritingZ =Z(GL(V )), it follows thatNGL(V )(F )/FZ ∼= CAut(F )(Z(F )), which is wellknown to be isomorphic to eitherSp2m(s) or O±

2m(2) (with s = 2 in the lattercase).

WriteN =NGL(V )(F ). We shall show thatb(N) is bounded. For this purposewe may assume that dimV = sm is large. First, by [14],b(F ) is bounded, so wecan find vectorsv1, . . . , vb (b= b(F )) such that ifJ =Nv1,...,vb thenJ ∩F = 1.

If J has a regular orbit onV thenb(N) b+ 1; so assume from now on thatJ has no regular orbit onV . This means that

V =⋃h∈J (

CV (h). (∗)

Let h ∈ J (. By [6, Theorem 2], there are 8(2m + 1) N -conjugates ofh, suchthat if K is the group they generate thenKF/F containsSp2m(r) or Ω±

2m(2).Using [7, 5.3.9, 5.3.10], we deduce that dim[V,K] (sm − 1)/2, and hencedim[V,h] (sm − 1)/(16(2m+ 1)). Since dimCV (h) = dimV − dim[V,h], itfollows from (∗) that

|V | |H |.qsm−((sm−1)/(16(2m+1))),

and hence|H |> qsm/(17(2m+1)). However,

|H | (q − 1)s1+2m∣∣Sp2m(s)

∣∣< (q − 1)s2m2+3m+1,

so this is possible only for finitely many values ofs andm. This completes theproof.

To conclude the section here is a lemma on the base sizes of classical andalternating groups on natural modules.

Lemma 3.7. Let Cld(pe,pf ) GLdf (p) and (Altm,pa) < GL(m−δ(p,m))a(p)be as defined in Section1. Then

(i) b(Cld (pe,pf )) 3de/f + 5, and(ii) b(Altm,pa) (logp m)/a + 5.


Proof. (i) Observe thatCld(pe,pf ) (F∗pfGLd(p

e)).f . Using Lemmas 3.1and 3.2, it follows that

b(Cld

(pe,pf

)) b

(F∗pfGLd

(pe

)) + 1 b(GLd

(pe

)) + 2.

By Corollary 3.4,b(GLd(pe)) 3de/f + 3. The conclusion follows.(ii) Recall that(Altm,pa) (F∗

pa Symm).a. As in part (i) we have

b(Altm,p

a) b

(F∗pa Symm

) + 1 b(Symm)+ 2,

where Symm acts on its natural moduleM over Fpa . Recall that this moduleis a section of codimension 1 or 2 of the permutation module of Symm in itsnatural action overFpa . Regarding the elements ofM as vectors of lengthm withcoordinates inFpa , we construct a base for Symm as follows. Divide1, . . . ,minto pa subsets of roughly equal size, indexed by the elements ofFpa , and letv1 ∈ Fmpa be a vector such that, forα ∈ Fpa , α lies in the coordinates correspondingto the subset indexed byα. It is easy to see that we may constructv1 ∈M in thisway such that the size of any two subsets in the partition differs by at most 3.To constructv2 ∈ M we divide each subset in the first partition topa subsetsof roughly equal size, indexed by the field elements, and as before we put eachfield element in the coordinates lying in the subset indexed by it. We go on andconstructv1, v2, . . . , vi until we reach a stagei where all the subsets created havesize 0 or 1. It is easy to see that we end up with a base for Symm: indeed, thestabiliser ofv1 is a product of symmetric groups of degrees corresponding tothe sizes of the subsets in the first partition (with multiplicities), the stabiliser ofv1, v2 is a product of symmetric groups of degrees corresponding to the sizes ofthe subsets in the second partitions, and so on, hence the stabiliser ofv1, v2, . . . , viis trivial. Finally, one easily sees that, by constructing subsets of sizes as equal aspossible, we obtain

i logm

log(pa)+ 3 = logp m

a+ 3.

The result follows.

4. Proof of Theorem 2

Suppose thatV = Vd(p) and thatH GL(V ) is irreducible and primitiveonV , with b∗(H) > C. By [14],H is non-solvable. We aim to show thatH is asin part (i) of the conclusion of Theorem 2. The proof proceeds by induction ond = dimV .

Assume now that there is a tensor decompositionV = V1 ⊗V2 of V (overFp)such thatH GL(V1)⊗GL(V2) and dimVi > 1 for i = 1,2. WriteHi =HVi ,so thatH H1 ⊗ H2. By induction, for eachi, eitherb∗(Hi) C or Hi is asin (i) of Theorem 2. Ifb∗(Hi) C for somei, say for i = 1, it follows that


conclusion (i) holds, using the fact that ifb∗(H0) C then by Lemma 3.3(ii) wehaveb∗(H1⊗H0) C. Thus we may assume thatb∗(Hi) > C for i = 1,2. Henceby induction, we have

H1 H0 ⊗s⊗1

(Altmi ,p

ai) ⊗

t⊗1

Cldi(pei ,pfi

),

H2 H ′0 ⊗

s ′⊗1

(Altm′

i, pa

′i) ⊗

t ′⊗1

Cld ′i

(pe

′i , pf

′i),

wheres + t 1, s′ + t ′ 1 andb∗(H0) C,b∗(H ′0) C. If di = d ′

j for somei, j , saydi = d ′

j = c, then

Clc(pei ,pfi

) ⊗ Clc(pe′j , pf

′j)

Γ Lc(pfif

′j) ⊗ Γ Lc

(pfif

′j)

< GLc2fif ′j(p),

and hence by Lemma 3.5,b∗(Clc(pei ,pfi ) ⊗ Clc(pe′j , pf

′j )) < C. Hence by

Lemma 3.3(ii) we have

b∗(H0 ⊗H ′0 ⊗ Clc

(pei ,pfi

) ⊗ Clc(pe′j , pf

′j))

C.

It follows that we may assume that all the integersdi, d ′j are distinct, and by

the same argument that all themi,m′j are distinct. Thus the conclusion of (i) of

Theorem 2 holds.We may therefore assume from now on thatH preserves no non-trivial tensor

decompositionV = V1 ⊗ V2 (overFp).Let K be a non-trivial normal subgroup ofH . As H is primitive, Clifford’s

theorem implies thatV ↓ K is homogeneous. IfV ↓ K is reducible then thereis a tensor product decompositionV = W ⊗ X such thatK GL(W) ⊗ 1X,K induces an irreducible group onW , and dimX = x > 1. Writing Fpe =EndK(W), we see thatCGL(V )(K) ∼= GLx(pe) acting homogeneously onV ∼=Vd/e(p

e) (see, for example, the proof of [7, 4.4.3]). AsH normalisesCGL(V )(K)and preserves no non-trivial tensor product decomposition ofV , it follows thatx = d/e; and soK F∗

pe is cyclic.Thus we have shown that every normal subgroup ofH is either cyclic or

irreducible onV . In particular, since the Fitting subgroupF(H) is not irreducibleby Lemma 3.6 (recall thatb∗(H) > C C1), F(H) is cyclic. SinceH is non-solvable, it follows thatE(H) = 1. HenceE(H) is irreducible onV . Let Fpf

be the field EndE(H)(V ), and regardV as the spaceVd/f (pf ); thenE(H) isabsolutely irreducible onV . Write

E(H)= S1 · · ·Sk,


a commuting product of quasisimple groupsSi . The spaceV has a correspondingtensor product decompositionV = V1 ⊗ · · · ⊗ Vk, where eachVi is an absolutelyirreducibleFpf Si -module.

SinceH preserves no non-trivial tensor product decomposition ofV , it mustpermute the subgroupsSi transitively by conjugation; hence dimVi is constant,say equal tom, andH lies in (

⊗k1GL(Vi)).Sk . If k > 1 we then haveb(H) 4

by Lemma 3.5, a contradiction. Thusk = 1. Sinceb∗(H) > C C2, it followsfrom Proposition 2.2 thatE(H)= Cld/f (pe) or Altm (with d/f =m− δ(p, d/f )in the latter case), whereFpe is a subfield ofFpf . HenceH = Cld/f (pe,pf ) or(Altm,pf ), and conclusion (i) of Theorem 2 holds.

This completes the proof of Theorem 2(i).Now we prove part (ii) of Theorem 2. LetH GLd(p) be primitive and

irreducible withb∗(H) > C, so that by part (i) we have

H H0 ⊗s⊗1

(Altmi ,p

ai) ⊗

t⊗1

Cldi(pei ,pfi

),

with s + t 1, H0 GLd0(p) andb∗(H0) C; also we takem1 < · · · < ms ,d1< · · ·< dt .

The proof proceeds by induction ons+ t . For the base cases+ t = 1, we haveH H0 ⊗M, whereM = Cld1(p

e1,pf1) or (Altm1,pa1). Write m = d1f1 or

(m1 − δ(p,m1))a1, respectively; so thatm is theFp-dimension in whichM acts,andd = d0m. By Lemma 3.7, we haveb(M) 3d1e1/f1+5 or 5+(logp m1)/a1,respectively.

Observe first thatd0 m; since otherwise, asm b∗(H0), Lemma 3.3(iv)implies thatb(H0 ⊗M) 3, a contradiction. Moreover, also by Lemma 3.3(iv)we haved0< b

∗(M). Hence Lemma 3.3(iii) gives

b(H) b(GLd0(p)⊗M

) 3

(1+ b∗(M)

d0

).

WhenM = Cld1(pe1,pf1), this gives

b(H) 3

(1+ 3d1e1

d0f1+ 6

d0

) 9

e1d21

d+ 21,

as required for Theorem 2(ii). And whenM = (Altm1,pa1), we obtain

b(H) 3

(1+ 5a1 + logp m1

a1d0

) 18+ 3m1 logp m1

d,

again as required. This completes the case wheres + t = 1.Now assumes + t 2. Let m be the maximum of the dimensionsdt and

(ms − δ(p,ms)), and writeM for the corresponding groupCldt (pet , pft ) or

(Altms ,pas ). Setr = ft oras , so thatmr is theFp-dimension in whichM acts. Let

N be the tensor product ofH0 and the other factors(Altmi ,pai ),Cldi (p

ei ,pfi );


soH N ⊗M. If b∗(N) C then the conclusion follows from the previousparagraph, so assume thatb∗(N) > C.

Letm′ be the largest among the dimensionsdi,mi − δ(p,mi) omittingm, andwriteN1 for the corresponding groupCldi (p

ei ,pfi ) or (Altmi ,pai ). By induction

we haveb(N) 9eid2i mr/d + 21 or 3(mrmi logp mi)/d + 18, respectively.

Consider first the case whereN1 = Cldi (pei ,pfi ). Sincediei d/mr, we have

b(N) 9di + 21. Ifmr d/mr then Lemma 3.3(iii, iv) implies that

b(H) 3

(1+ 9di + 21

mr

).

Sincem di and alsom > 21 (otherwise an easy argument using Lemma 3.3givesb∗(H) < C), this yieldsb∗(H) < 33 C, a contradiction. Thereforemr >d/mr, and now the conclusion follows by the argument given for thes + t = 1case.

Finally, if N1 = (Altmi ,pai ) then the fact thatd mrmi implies thatb(N)

3 logp mi + 18, and the argument of the previous paragraph gives the conclusion.This completes the proof of Theorem 2.

Deduction of Theorem 1. It remains to deduce Theorem 1 from Theorem 2.Suppose thatH is a primitive irreducible subgroup ofGL(V ) = GLn(pe) withb(H) > C. Then F∗

peH < GLn(pe) GLd(p) with d = ne, and F∗

peH isprimitive and irreducible onVd(p). ReplaceH by F∗

peH . ThenH is as in (i)of Theorem 2.

Assume first thatms dt . NowH containsCldt (pet ), and the order formulae

for quasisimple groups show that∣∣Cldt

(pet

)∣∣> 1

αpet (d

2t −dt )/2 whereα = 2

(4,p2 − 1

),

whence

log|H |log|V | >

et(d2t − dt

)2d

− logp α

d.

By Theorem 2(ii) we haveb(H) 9etd2t /d + 21, from which it follows that

b(H) < 18log|H |log|V | + 9etdt

d+ 21+ 18 logp α

d.

If etdt < d thenetdt/d 1/2, and sinced > C 33 it follows that conclusion (ii)of Theorem 1 holds. And ifetdt = d then Lemma 3.7 implies thatb(H) 3dt/et + 5 (in particular, the constant 9 in Theorem 2(ii) gets replaced by 3),and Theorem 1(ii) holds again (indeed, a much stronger bound holds in this case).

Suppose finally thatms > dt . ThenH contains Altms , and using the inequalityn!/2 nn/2 for n > 4 we obtain

log|H |log|V | >

ms logp ms2d

,


providedms > 4. By Theorem 2(ii) we have

b(H)3ms logp ms

d+ 18,

and the conclusion of Theorem 1(ii) follows easily as above.This completes the deduction of Theorem 1.

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