on multiply transitive groups

On Multiply Transitive GroupsAuthor(s): W. A. ManningSource: Transactions of the American Mathematical Society, Vol. 7, No. 4 (Oct., 1906), pp. 499-508Published by: American Mathematical SocietyStable URL: http://www.jstor.org/stable/1986241 .

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ON MULTIPLY TRANSITIVE GROUPS*

BY

W. A. MANNING

The properties of primitive groups that contain transitive subgroups of lower degree were first investigated by JORDAN.t In the Traite des Substitutions he proved that if a pritnitive group of degree nt contains a circular substitution of prime degree (pv) it is at least n - p + 1 times transitive. The capital importance of this theorem led him to examine the general and much more difficult case, that in which the subgroup of lower degree is merely transiti ve. He obtained the remarkable theorem: t

"' If a primitive group G of degree n contains a group F, the substitutions of which displace only 1- letters and permute them transitively (p being any integer), it is at least n -p - 2q + 3 times transitive, q being the greatest divisor of p such that we can arrange the letters of F in two different ways in systems of q letters which have the property that each substitution of r replaces the letters of each system by those of a single system. If none of the divisors of p have this property (which will happen notably if F is primitive, or formed of the powers of the same circular substitution) G is n - p + 1 times transitive."

NETTO ? and RUDIO 11 later gave proofs for that special case in which F is primitive. The only other contribution to this theory was made by MARGGRAFF.?

He proved that if q is the greatest divisor of p such that the letters of r may be arranged in systems of imprimitivity in at least three different ways, and if p is divisible by some number r such that F admits r + 1 systems of imprimitivity with one letter in common and no two of which have more than one letter in common, G is at least n - p - 2q + 3 times transitive. If these two conditions are not fulfilled, G is n - p + 1 times transitive. In MARGGRAFF'S

*Presented to the Society (San Francisco), February 24, 1906. Received for publication June 19, 1906.

t C. JORDAN, Traite des Substitutions, 1870, p. 664. 4C. JORDAN, Journal de Mathematiques, ser. 2, vol. 16 (1871), pp. 383-408. ? NETTO, Journal fur Matheinatik, vol. 83 (1877), pp. 43-56. RUDIO, Journal fur Mathematik, vol. 102 (1888), pp. 1-9.

T MARGGRAFF, Dissertation, Ueber primitive Gruppen mit transitiven Untergruppeu geringeren Grades, Giessen, 1889; and also, Wtssenschaftliche Beil(age zum Jahresberichte des Sophien-Gymna- siums zu Ber-lin, 1895, Prograinm nr. 65.

499



500 W. A. MANNING: ON MULTIPLY TRANSrrIVE GROUPS [October

second paper are given some interesting relations between n and p, for example, p2 9.

The word " two " in JORDAN'S theorem, and " three " in MARGGRAFF'S may in all cases be replaced by " q + 1."

Let it be assumed that G is a doubly (and not triply) transitive group of degree I: rXI ( qr= 1), in which are found transitive subgroups H' (i= 1 2 .., r) of degrees =l, qx respectively. It is also assumed that the degree of any transitive subgroup of G which displaces more than q, and less than E', qx letters is one of the numbers Z=, qx ( i =2, 3, r, -1). This does not say that there is not a transitive subgroup of degree less thani q,. Let a> (k 1, 2, ., q,) be the letters displaced by Hli but not by Hi-', and let 174 be the largest subgroup of G on the letters of Hn; IF' includes all the preceding subgroups and is transitive.

If qr= 1, G is triply transitive contrary to hypothesis: hence qr 1. If for those values of i corresponding to which qi > 1, the letters a, (k = 1 2, q do not form a system of imprimitivity in F1, some substituition S of IF' will give a group S-'Pj'S displacing some of the letters a',, (k= 1, 2,.., qi) and leaving fixed at least one of them. Then { F->, s-'ri-F S} is a transitive group of degree greater than Z> 'q and less than qi= x , contrary to hypothesis. By the definition of imprilnitivity all the subgroups of 174, nlotably F1, Fl, ., r-' , admit these systems. It should also be retnarked that the letters (tx k (k = 1, 2, ... , qx) fall into sets of imprimitivity q, by qi, for x =1, 2, *,-1. Since qr> 1, then qi> 1 (i = 1, ,, r -1).

Let ar 1 represent the system ar k (k = 1,2, . ,r) of [r, and let a" (i- 1, 2, . .., r-1; k=1, 2, .'., 7/iqr) be the remaining systems into which a> goes under the substitutions of rF. We define 174 as the group in the systems a (x=1, 2, ..., i; k=- 1, 2,., 777r) of 174. Obviously I' is at least doubly transitive.

The group G, being doubly transitive, containis a substitution 174 = (ar+ a7

r Let it be granted for the moment that T17 transforms F , Fi+2, .

2 ,r '-1 each into itself and leaves fixed the letters a'+', a>, i *2 a r-I

Then { ]P 17Til, } =l P; for, its degree is less than + _i q. and must con- sequently be E2=> qx, that of r,i. If 174 displaces a> 1 there is a substitution S in FU which replaces a> 1 by the same letter as does T7i. Then S-' 17i leaves fixed al. 1as well as ai-1, ... al 1 and may be used for 17T. But it is clear that T17 (al, Ila>,) . . transfortms I' into itself, and by mtultiplicationi by a substitution of r>r-' a like substitution may be obtained which leaves a-' fixed. Then by a complete induction G contains a substitution

fori =1, 2,I.).(a.,,) w r m g P)(alhm P Pl, l .* i

for i, 1, 2 .,..r wbich transformis each group ri, ir, , ]r,-1 into itself.



1906] W. A. MANNING: ON MULTIPLY TRANSITIVE GROUPS 501

Each system of letters a, k (k = 1, 2, .., q^) is transformed into itself. The symbol a3 r- will be introduced later for the system a (k = 1, 2, *.,

and other properties of T1 pointed out. Now Ti-j ]P'-' =l Frl- ( 1,2, .1 . ., qr), but the system a2j 1 is replaced

by q, other systems which have at least the letter a] ' in common with ar2-1. Let these q,. new systems of imprimitivity of r'-1 be represented by a2-?* If it is assumed that aY r-I has a second letter a r-I in commonl with a r-1, Til has the sequence a -1 a-j, where al is some letter of the system a2 . In 2 there is a substituition S = (ar2- ar l).... Now US= 5-'Til S leaves fixed the letter of ar.) which follows arl-1 in S, replaces the letter which follows a-j in S by the one which follows al rI, and certainly replaces a letter of a rj by ar+1'. Hence one sees that the degree of { U-1 Frl- U, Fr-I } is greater than Er-' qx and less than r=l qx, contrary to hypothesis. The q, systems a"-'.l have one and only one letter in common with a.r-1

Suppose it possible to find in G a substitution T that transformns ]rl- illto

itself, and which replaces the system a r-> l, by a system a' distinct from any system resulting from the qr + 1 arrangements obtained above. If a' does not include the letter a>, some substitution S" of F>r-1 will replace a' by a system a which does include a-'. Now T,1 TS" replaces the system a>r- by a and the letter a>r-1 by a> say. We take from r-1 a substitution SI which replaces a> by a>, and in consequence leaves a fixed; the produict U- Ti TS"S' leaves a> fixed and replaces a>-' by a, so that U= (ar1>)(a+ al r

Now UTj =_ (a,,>1 ) ( a,) ) = S, a substitution of rF, and therefore U= ST, k C qr. Hence a, the result of transformning Fr>- first by S and then by T,, is merely one of the systems aryj' -

The system arjjl , bears the same relation to rr-1, T7 lrrTil, and G, that

ar>1 does to r1-, r1, and G. From ari then cani be obtained, by nieans of substitutions that leave a>-I fixed and rr-1 invariant, q, other systems with but one letter in common with ar-1. But we have just seen that these q, systems will coincide with systems already obtained. Hence the letters qf Fr>- may be arranged in systems of imprimitivity of q, letters each in at least qr + 1 ways with one letter common to the qr + 1 systems and with no other letter common to any two of them. This theorem holds 'a fortiori for all transitive subgroups of r-1, in particular for RI'. It was first proved by MARGGRAFF, Ioc. cit.

Since Ti, permutes the letters al- (k = 1, 2, ., qr-1) among themselves, the letter s (q2 in number) involved in the systems a"<, az 1, a a12* ,, 1.

are all found in the larger system a>-' above. Hence we have the inportant relation q, _ q2.

It will now be shown that rr is iot in general triply transitive. An excep- tion arises only when r = 2. Suppose I'2 triply transitive. To each of the q,+1 distinet arrangements of the letters of 'r>- in systems of imprimitivity of



502 W. A. MANNING: ON MULTIPLY TRANSITIVE GROUPS [October

q, letters each, corresponds a doubly transitive group according to which systems are permuted.* Let ?7' (acj) be the subgroup of IF' that leaves fixed the letter a711. The remaining q - 1 letters of each system of imprimitivity to which (aWr1 belongs may be permuted only ainong themselves by the substitutions of rr-I ( arj). Now r'-1 ( ar-j) is transitive of degree ( l /qr) Ez4 qx 1 , while the number of systems transitively connected by it cannot exceed qr - 1. Hence Ex=_ qr Cq, anld (since qr-1 - ) qr- = q= q and r =2, unless r?' is a regular group, in which case also r = 2. It will be shown later that an imprimitive group of degree q2 in which q + 1 systems of q letters each with one letter in conmnmon, and such that no two of these systems have more than one letter in commoni, is either regular or of class q2 _ 1. However, the proof of the theorem we are going to establish is already coitmplete for r = 2, whether 11 is triply transitive or not. In completing the proof we assume then that r > 2, so that F" is doubly but not triply transitive. It is such a group as G itself and to it may be applied all precedirng results for G.

Is it possible for F1 to have a transitive subgroup of degree greater than

ql/lr and nlot equal to one of the numbers (l /qr) ZEi=l qx 2, 3, ..., r)? Let, if possible, rT, kbe a transitive subgroup of rT+j which displaces besides the letters of ri the k letters a'+', a7+21, a* *,'+(1-k qj+jq r) Then between rP and ri+1 there is a corresponiding subgroup r k, which includes ri and is transitive in the systems a' . , so that rf k certainly has a transitive constituent of degree El=t qx + kqr Now transform rF by all the substitutions of pl k* If the group generated by r' and all these transforms is not tran si- tive of degree E qx + kqr, it is because kqr E=iqx an absurdity since kqr <i+il,+, qi, that is, kqr < q,. This holds for i = 1,2,., r -2. Nothing is said about possible transitive subgroups of various degrees in rF.

We may now form the doubly transitive group in the systems of r?'. The system al- j (k = 1 2 , q-1Iq) will be represented by a r-I. The large systenm (t2, k = 1( , 2, *, qjqr) breaks up into the smaller systems a", (k= 1,2, *2 qIq-,) This group on the letters a-3 ' (i-i 2 * .,r-1) will be indicated by rP7, and the subgroup of it which corresponds to rP is ?. In the same way we proceed to form the successive groups. r in the sys- 3 .1~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

*Cf. C. JORDAN, Journal de M athematiques, ser. 2, vol. 16 (1871), third paragraph of article 7 on page 388. From the fact that " I" is doubly transitive with respect to the systems of q letters s, sl, * * - which it contains, it does not follow that if its letters can be grouped in different ways in system-s of imprimitivity, each of the new systems will be contained completely within one of the systems s I , ***. For example it is not true in the regular non-cyclic group of order 6. In it systems of two letters each are permuted according to a doubly transitive group, while there are three distinct ways of arranging the letters of the group in systems of imprimnitivity of two letters each. But the argument is valid if use is made of the fact that "II" contains substitutions leaving fixed at least two letters. To see how the subgroup " H " of " I" should be brought in to complete the proof, see MARGGRAFF, 1. c., theorem X (1889), p. 20, and (1895), p. 16.




tems a" k (k_ 1,2 , iqj/qr<j+2; i,j= 1, 2, ...,r, provided i +j r +2; and q9+, = 1. The groups FJ for which i +j = r + 2 are all doubly and not triply transitive except perhaps 11. From 172i>+ we have

q. /q.\2 2

q,'-I or ( q2 )I (i=2, 3, ... r),

so that r-i~ ~ ~~~~-i1(- +11

q~~q~?1 qi+ I, or qi =qr or qi qi+l I qi+ ir qi-/1o q(rl)/

The operator T,1 = (a j' a ) .. we have seen, leaves fixed the letters al 2 qr1 r- r2 -3 2 1 a2 l *, ala , and the systems aC j, a4-i2 ar,1 * * 9 ar 1, arl+l 1 Since Ti

leaves fixed the system arj+2 of rP-j'+2 (j _3 4, . r), it permutes the systems a' (k=1, 2, 9 qi/qr-j+2 ; i=1 ,2, -.,r-j+ 2 among themselves, without a change to an arrangemenit essentially distinct. Then Til leaves fixed all the systems al (j=1, 2, ..., r + 1; i= 1, 2, , r-1, provided i+ j r + 2, and, when j=2, i r-1 ). In exactly the same way we get a suibstitution T1 = (a57i+2 a?l ) ... which leaves fixed all those elements which bear to r-j+2 the same relation as the elements left fixed by Til bear to G. If we consider the group I-" it is clear that a substitution S can always be found in it that replaces a certain letter a' by a,I and that has the property of leaving fixed each of the letters a' ,, 1 , a- 1 , and hence also the systems aIl ,t2,, , a+-, I1, al l y With the aid of such substitutions we may choose L. so that it leaves fixed all the elements a' 1 (y = 1, 2, , r +l; i = 1- 2 , r j,provided y + i c r + 2, and i + r-j when yj). It is clear that Tl. transforms rF ( i = 1, 2, , r -j) and 'r-j+2 each iInto itself. We should bear in mind that ]Tf may also be regarded as a substitution of ]r-i+2 and leaves fixed all the letters a,k (k = 1, 2, *, qi; i = r-j + 3, , r + 1

The following theorem will now be proved by a complete induction: The letters qf F>-' may be grouped q, by q, in systems qf imprimnitivity

which have an arbitrary letter a'-' in common in at least qi.r?1 l/q, distinct ways. Furthernmore, (1) A-ny two qf these systems have exactly qi+l letters in

common. (2) The letters involved in these systens are all included in the larger system a'-'+3 . (3) No substitution of' G which leaves F>' invariant can replace one of these systems by a system, 2f imprimnitivity not included am,onq those into which one of them goes by a substitution of P"-' itsetf. (4) Tlkose systems which have a'-' in common preserve the orififlal systems. all, (5) The q, systems of q, letters which htave a'-' but not a'-' in comnmon contain no letter of the systenmt a'-j except a'-.

We assume the truth of the above theorem for the subgroup r1 and its systems of q,,1 letters, and shall show that it must then hold as stated for Fj'. Now if it holds for I" it holds also for fi-l. Hence systems of qjq, letters 1 2




with a-' in common may be chosen in qi r ll/q. distinct ways, q,iEr 11/q of which have the systemn a'-' in common. Now transform I>-' by

Til (il = 1, 2, , q.). Since Til does not replace the systems a',, of F[-' by other systems essentially distinct, the q E- 1 lqx systems with ai-1 in common undergo no change. But since Til leaves fixed aC'-1 and may replace letters of ai-1l only by letters of a'-', and since a'-'l is the only letter of a'-' in the remaining qi/qr systems, no two of the block of ( qi/qr) (1 + qr ) systems obtained by this transformationi are the same. Since Til leaves fixed the system a'-'I+? (the system of qi letters from which we start), each system of the total number,

qi Ev l/q ? q,(l + qr)/q== q Zybjl/q., has just qil letters in common with -' . Obviously all the letters of all these systems are included in a -L+ The lnext point to establish is that no substitution of G under which F1-j is invariant can replace any of the qiEr r l/qX systems with ar,-' in common (the system a say) by a system (at") not found among the Er' qr * 1 /Iq

systems of I"-' resulting fronm the above systems which have a'-' in common, after transformation by all the substitutions of Fl-1. By hypothesis all substi- tutiolns of rr under which r'-1 is invariant give rise to no new systems. Let T be a suibstitution of G which replaces the system a by a", and let T-lrl-FT7 F-1. Let Til be that substitution (defined as before) which replaces one of the systems of q, letters which include aji1, by a. Since TiRT cannot be a suibstitution of rr , it must replace ar+ by some letter a' k (i<r+1), The group { TI, Ti, Tii,_ } leaves rJ-' invariant, ai-1 and a'-' fixed, and has a constituent which is transitive in the letters a-] (k = 1, 2, . qi;

= i, i + 1, r, r), so that from it we may take a substitxition S"' which replaces the letter a", that follows aj1 in Til T by a letter a, i. Now T i1TS"' replaces a'r+I by ar 4; let a' be the system into whieh S"' changes ac". Since S"' is a substitution of J'1, ' satisfies the condition imposed on a". Let S" be a substitution of '-1 that replaces the system a-' by another (ca) which includes the letter ai-'. Finally let S' be a substitution of rF-' the inverse of which replaces a'-' by the same letter as does TilTS"'S"; S' leaves fixed the system ac. Then U= TYillTS"' S"S' leaves rF-' invariant, a'-' fixed, replaces a4r+1 by al r, and changes one of the systems which have aj-' in common directly into ca. Now UuT,1,= (a'-j)( ajr+1). .. =S, a substitution of rr, which leaves the ele- ment a'-' fixed since ai-ij, one of the letters of a'-, is fixed. From U= ST7 it is clear that aC must be one of the set of systems obtained from those of I-' by means of iT,.

To prove that anly two of the q, 5it_ 1 /q, systems with one letter in common have qi+, and only q,+, in commion, we have only to note that the series of transitive subgroups F" (x = i, i + 1, *.., r) may be so chosen that any one of the systeins involving the letter a'-' may be made to take the place of a"-I+2 l- In fact { Til Ti2 . Ti.,_, } contains a substitution T which replaces any one




of the q.^,+' l/qZ systemns which include ah-' by any other and permutes them only among themselves. Such a series of subgroups is Ir , 1, . .., j-], T-'FI T, T-1'F"T, G, and in it an arbitrary system plays the part of i_I

arx+2, 1&

It remains to be shown that the qi systems which have ai'- but not a'-' in comnmon contain no letter of the system aj-,' except a'-j. These q, systemiis are those by which the q/lq,. systems in the letters of r7-', which do inot have a'3-' in common, are replaced by Til(il = 1, 2, . qr). By hypothesis these qi/qr systemns contain lno letter of the system a'-1, except a'-', and Til permutes the letters a'-j', , of a-1', among themselves, giving in place of the system a,j, q, systems which have no letter in common with it except a'-'. Hence none of these (qi/qr) q, = qi systems have, besides ai'-1 any letter in common with a'.-

This theorem has been proved for the systems of q, letters in ]r-,, and in consequence holds for the systems of qr_1 letters in rr-2, and so on. In regard to H' we may now say that systems of imprimnitivity qf q, letters may be chosen in q . 1 /q, distinct ways, for i = 2, 3, ., r - 1, r.

If G is contained in a larger primitive group C' of degree n, G' is n- E"=, + 1 times transitive. Now

q ql + + T + + JrI) 2qI (1 - q) -2qi(i 1

Hernce the theorem,: A plimnitive group o degree n wphich contains a transitive subgroutp o

degree q1 is (t least n - 2q, + 3 times transitive. We may give another incomiiplete but useful statement of our theorem: If a primitive group G of degree n contains a transitive subgroup II qf

degree ql, G is at least n -l qr( q2 )/(r 1) + 1times transitive, q2 being the greatest divisor- of q, such that H has at least q2 ? l distinct arrangements of its letters in systems of iinprimitivity of q2 each, and qr being the least d'ivisor of q, such that HI has at least qr + 1 systems of qr letters each with one letter in common, and not more than the one letter common to any two of them.

I.f both these conditions are not satisfied by the given group H, G is att least i - q1 + 1 timbes transitive.

From qi_l-q,/q,+,, it follows that qr-+l c q 2 _ (r-i+1)(r-1) Then

r-2 r-3q2Iq l q3 + q4 + + qr q2 + q2 + + q2 1/(r-1) 1

Ilence G, which is n1 - r+1 q, times transitive, is at least




n - -Ql 2 + 2 n n l q2 - _ -1+2r q~ q - +1

times transitive. This limit is attained by the holomorph of the Abelian group of degree 2a

and type (1, 1, ...), which is triply transitive and has a transitive subgroup of degree 2a-1, for which q2- 2a2 and qr= 2.

Other useful conditions which F'-j must satisfy may be obtained by consid- ering the meaning of the multiple imnprimitivity it exhibits.

Let J be an imprimitive group of degree n and order nm; let al, a2, * a.

be the letters of a certain system of imprimitivity of J. The suLbstitutions, qrn in number, that replace al by a,1, a 2, .., aq respectively, leave the systemn a1, ..., aq fixed. The product of any two of them also has this property. No other suibstitution of J permutes the letters a, ... ., aq among themselves. Then these qm, substitutions form a group (H). Let there be another system of q letters in J which has the letters a1, a2, ..., aa in common with the first, anid the reinaining letters a'+,, a'+?, . , a' distiniet. The group If' which corresponds to this systenm has exactly qa substitutions in common with H: those which replace al by a1, a2, , aa These amn common substitutions forim by themselves a group (F) .

Now if a substitution T transforms J into itself, permutes the letters a2, a3, *--, a, among themselves, leaves 1, fixed, and replaces aa+1, .., 9q by aa?,, .., aq, it transforms F into itself and H into H'. We conclude that Fj' has a set ?f qiE r I I/q_ subgroups of order qinmil which have in common the subgroup (of order rmi-) of F"' that leaves one letter fixed. Any two of these subgroups have in common also a subgroup of order qi+l? i-. This set of qi. j I 1 /qx subgroups of order qi mIni must be such that some iso- niorphism of rF1j to itself brings any two we please into a 1, 1 correspondence. In particular, no one of them can be a characteristic subgroup of F-j'. If one is invariant in rl-. all are invariant.

We recall that rr-1 admiits q + 1 arrangements of its letters in systemis of imprimitivity of q, each such that q. + 1 systems may be taken with one common letter and no two of which have more than this one letter in common. If al be taken as the leading letter, the q - 1 letters thus associated with it are all found in the larger system a3,. It may happen, as when qr-l - q2 that the

qr letters in question forml a systern (A ) of imprimlitivity of FV'. The subgroup A' which leaves this system A fixed has a transitive constituLent on the q2 letters of A. We call this constituent lroup A and suppose all the otlher letters of A' erased. Now A has the properties of Pj-1 in so far as systems of imprimitivity are concerned -even more extensive properties perhaps. Conl- sider a substitution S of A which leaves fixed two letters aj and ; Let S=(al ap, *p)a * Since S can only permute among themselves the letters




of each of the systems to which a,j belongs -and similarly for a"j-'- the letters arp' and al- belong to that system which contains both al rj and a"-' Then S displaces at most most q r- 2 letters, from which it readily follows that S is the identity. We conclude that when A is not regular, it is of class q -1 and has a characteristic transitive subgroup (e) which is regular.* This regular subgroup of order q2 occurs in both cases, anid A, when regular, may be called e for the sake of uniformity. This group e admits a (qr + 1) fold division into systems of imprimitivity, no two systems having inore thall one letter in common, hence its substitutions are distributed amtlong q, + I subgroups of order qr, no two of which have an operator in common other than the identity.

Let s1 =, S2 ** , Sq be the substitutions of one of these subgroups, one of whose substitutions is conjugate to sorile substitution t. not in it. Let

tI 1, t2, 9 , tj, *.*, tq7 be that one of the qr + 1 subgroups in question which includes tj. Now every substitution of the group e is given by

sato( a, 3 =1, 2, , q,). But clearly t 1s- sis,s,to + tj. Then each of the q, + 1 subgroups is invariant, and since no two have anything in common but the identity, every substitution of one of these subgroups is commutative with all the substitutions not in it. Then the group e is Abelian. Let qr Mpa, where, if possible, m is prime to p (p being a prime number). The Sylow subgroup of e of order p2a must have mpa + 1 subgroups of order pa

with nothing in common but the identity. Then

1 + ( mpa + 1 ) (pa 1 ) = p2a,

so that mn = 1. If now we take the two suibgroups of order qr = pa in which the subgroups composed of the operators of order p are of the lowest possible orders pkl and pk2, k2D we get the inequality pki+k2 a pk , whence ki 1=2 a= . Then e is of type ( 1, 1, ) t The elementary group of order 16 has 5 subgroups of order 4, no two of which have anything but the identity in common. It is found in the quintuply transitive group of degree 24 of MATHIEU.4

Another assumption we may make in regard to I"'1 is that there is in it ail imprimitive system of q2 + qr letters including al r1 anid the q2 _ 1 letters associated with al-j' in imprimitive systems of q, letters each. Just as in the preceding case there is a subgroup A' that has a transitive constituent A of degree q2 + qr which we now investigate.

Let S be a substitution of A that leaves fixed two letters a] and a-. By *FROBENIUS, Berliner Sitzungsberichte, 1901, pp. 1216-1230, and 1902, pp. 455-459. tCf. MILLER, Bulletin of the American Mathematical Society, vol. 12 (1906),

pp. 446-449. This is the proof Professor MILLER mentions in the note at the bottom of page 449. TMATHIEU, Journal de Mathe6matiques, ser. 2, vol. 18 (1873), pp. 25-46. MILLER, Bulletin de la Societe mathematique de France, vol. 28 (1900), pp.

266-267. Trans. Am. Math. Soc. 34



a50 8 W. A. MANNING: ON MULTIPLY TRANSITIVE GROUPS

the method used above we know that the only letters which S can displace are the q r - 2 other letters of a system which includes both acj and aljk, the qr letters not in a systemi with a"-1' and the qr not in a system with a, 3q, -2 at most. But if S ( aj . . *. ) a it must replace every system of q, letters in which ar-I is founld by a system in which al is present. Then S displaces at least q2-qr + 2 letters. Now q2 qr+ 2 > 3qr -2 when qr > 2. It is obvious that when qr 2, A is regular. Then for all values of qr, A is either regular or of class qr + 7r - 1. In the latter case A contains a regular characteristic subgroup. Then the regular subgroup e of order q2 + q, is always fouind in the constituenit A of A'. Let A1, A2, *, Aq,A be the subgroups with nothing in commion but the identity which correspond to the q, + I systems of imprimitivity of qr letters each in question. In ) systems of qr letters each are permuted according to a transitive group of degree qr + 1. All the substitutions which lie in the stibgroups A1, leave fixed at least one of these systems of impriinitivity. This is seen by applyinig to e all the substitutions of its coiljoin.* Then ( has just qr substitutions which displace all q, + 1 systems. The group of degree qr + 1 in the systeins is of class q,. For it is a characteristic property of groups of "sclass n -1 " that they have just n - 1 substitutions of degree n.t .Hence E4 has a characteristic subgroup HI which leaves fixed none of the (q, + 1)2 possible systems. It is clear that A1, . . are conjugate under H. Consequeintly the group in the systems is of order (q, + 1)qr and doubly transitive. Hence H is Abelian of type (1, 1, .. ) .t If

qr-I qr (qr + 1), we have q,-1 + qr -. 1 - p2a The doubly transitive group of order 432, degree 9 and class 6 is a case in point. The transitive suboroulp of deg-ree 6 in it is noni-cyclic.

STANFORD UNIVERSITY.

* JORDAN, Traite des Substituttions, 1870, p. 80. t BURNSIDE, Proceedings of the London Mathematical Society, vol. 32 (1900),

pp. 240-246. +JORDAN, Journal de Mathe'matiques, ser. 2, vol. 17 (1872), pp. 351-337; FRo-

BENIUS, 1. C.



on multiply transitive groups

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