Primitive permutation groups with a sharply 2 ...

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Primitive permutation groups with a sharply 2-transitive subconstituen of prime degree Jie Wang

*

a

a

Department of Math, Peking University, Beijing, 100871People's Republic of China Published online: 05 Jul 2007.

To cite this article: Jie Wang (1992): Primitive permutation groups with a sharply *

2-transitive subconstituen of prime degree , Communications in Algebra, 20:3, 923-941 To link to this article: http://dx.doi.org/10.1080/00927879208824382

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COMMUNICATIONS IN ALGEBRA, 2 0 ( 3 ) ,

923-941 ( 1 9 9 2 )

In this paper we have d e t e r ~ i n e ddl lmiprirnitiv? permutation groups wkicf; have a stxpl:; 2-tiansitive subconstituent of prime degree.

$1. INTRODUCTION Let G tie a uniprimitive pernutation group acting on a Szite ser 2 , f o r a 5 0: suppose G, h r s an o-bit A(aj and the corresponding t;%nsitive constituent

~ , h is~sharp!:? ~ '

2-transitive. Then by 1111 Chm!!ary 3.9, '6,

'

d ( d - !)"or

d = iA(a)i. In this pa.per, ive c!as~ifi.all such polips provided d = p is a prime. All sharply ?-transitive permutatinn groups were determined by Zassenhaus (211.

It follows that G'e'atk Z,

: ZP-,

so!rable. W e also assume that p

if ( ~ ( a . j=; p . And by [20: 16.3, G, itseif is

> .5, since all prinitire groups G r h an orbitz!

of length 2 or 3 have already beer1 determined. C. E. Praeger :I61 prored that, if G has a doubly transirive subconstituem,

then either G is almost simple, or G has a unique minirna.1 regular normal s&,rrrcip Jf C: has a regular norma! sukgrcup 10. then

I t fo!!ows t h a r L! E [ 6 , 5 . 7 ] . Eler?.ertar:; caicuia:ion shcn~stha: on!;; d = 3 ar?d q = 1 or i are possible. T = P S L f 3 , 4 ) has been inc!uded io Tab!e 1 whi!e

P S L ( 3 , ; ) is exchded 5y

:.j:.

PEOOF:In this ease; ill has h r s n y e primp r. So

.?f is

2

s.:k;rc:1p

gf

a sector isomorphic .co

PSL id,yo) where

Q

=

qi

is not solvable except for d = 2 and qo = ? c~r3. Hellce

5:.3 r ?.:,:.?Y,

If .!f 5x5

3

~ e c t c riscmorphic ~c Zp : Zg-I.

t h e 2 r = p. This i r p l i ~ sthat Z, is a, direct fzctcr n f M , a contradirtion.

PP,.OOF:in this case: ,if = =Yc(Ei) for W = P 5 ~ ( d , ~PSI:(d:q)(q j, = r2) or Pi?jcl,q)(qodd). ff H is symp!eetic, ao so!sable

can be found, If f-I is

. A L a y , then ( d , y') = ( 3 : 41, (2: 4) or ( 2 , Q ) . Hcwever, PSL("4) 2

I,c..-

and

P S L ( 2 . o) r .A6 hzve been treated in section 2: while F S L ( 3 , 4 ) is exclvded by

.a ' . Fisz!i~,if H is orthogonal, ( 2 : ~ =) 'cl


>

( 8 , 3 ) or j a ! i ) , both are excluded by

In this sulsec-,ion, suppose that T = P S p ( d , q ) ; ~ h e r id 2 1 is s e n . If d = 4 and q is wen: assume that G contains no g a p h automiorphisms. Moreover. !et I' be the d-dimensional s.;mpiectic q a c e over GF(q).

PgOOF:

In this case, M normalizes a schspace U of I', a n d either U is non-

degenerate

OF

t ~ t a l l yisotropic, In the former case, :V is unsolvafle. I n the

latter case? siippose t h a t dim(L7) =

PSL(r! q )

\:

c:ntradic:ion,

7.

T h e a M has a seceor i s o m ~ r p h i c-co

PSp(d - 2r! q ) , This implies that g = 2 , 3 a n d K(M)= f2:3 ) : a


SHARPLY 2 - T R A N S I T I V E PERMUTATION GROUPS

935

PROOF:If +Li tf C;, then .ti = ,YG(X) for X f S y 1 2 ( T ) and , X = 2".

Let

3 .ri c:ll,isuch t h a ~33,q,iK 1 Zy_: 2,-;for sgme prime p L 5 , then X K I K 5 Z, :

ZPd1 is a norma! 2-subgrijuy, i: fo!!ows char -1-5 K aud

4 r^

so 2--

I t f ~ l i o w sthat (q - 1)' , ( q - - I ) . j. IT is easy co ~e;ifj: i h ~ jqi 1

hence ( q - 112 ' f. This

0

meam that (41

The same argument shows that

5

-

!I2

,p

- i.

!:j?

TLis

- i) = I

j for f 2 2, which is impossible.

= - 1 is also impossible.

If :ti f C2:r h e a M = !YG(-Y) for -'Y

%

Z4r+l. \Iembers of this class ar:?

nbta,ined by the fusion of the original Cs arid C8 ~ e ~ b e rwhich s , are isomorphic


PROOF:T h e members of C2 contain PSC'(a.,q'')iS,, there%re r: 5 3 , ;::. r 5 4. If s > 1, then it is n o t difficuit to show thze r j M ) = {?:3). conrradicting the z.ssumptloc that M ha.s a sector isornorpkic t o 2, : Z,-l.

When r = 1. 'M (-?TI= q

+ 1 , buc jp

2 7 g ~ n l e xshoi::s

1)"

. 31

,

If s = I , chen r = 3 : 4 .

it follows ?h+t ,o am! he a cli~isorof

i.W. con~radiccingche facx that iMi i p j p - 1'1" The same

that iio coun:2r eszxple occ-ms xhen r = 4.

937

SHARPLY 2-TRANSITIVE PERMUTATLOW GROUPS

PROOF:In &is case, 3f has a sector isl~morphicco P ~ C ' j d ; ' f , ~fcr ? ' :some odd forces d = r . hence if C T Z Z, : Zd :%:here i: = prime r. i n e so!r.abIiity cf -7

r&. so $1

Suppose rhat q =

5 Z,

f: for some prime

: Zd.[S(d,q- 1: j l.Clear!:;

in Lemma 5.1.3 we can assume that


IS (d, q

l)

= I.

:hen

i. then O u t ( T 1 = 2 i d . y

-

l j/

( d : q - 1) = 1 or d as d = r is prime. A s 3'0,


PBOCF:If .9[ 5 Cj:

as .9f is s o l ~ . - a t l e , only :he 59!lo::-in.g

.t~~:c:cases haye

be

considered. '

( 1 ) T = PSC(3>?");r is an Paa prim* nuEber, and .!.ifiT G PGU(3,2?j. 1.

In this case, lOut(T\' = {3: 2* - 1 ) r . If there esists a normal subg-oup K such &at i\fjh'

1 2, : Z p - : , and

(2) T = PSC;;, q$), 51 fl T

M

'ii; p - I , chcn p = r, Meariivhile: _ti has a

nnr~t:~! subgroup 9'such tba; :'rP.'iY E Z,. This implies J d 'Ei 2 Z,

. . contrad!cr~c:n.

c

:, Z,-!.

a

PQ(d, where 4.0 is a n odd number. The snivr?bllit:;: cf ZI !.ads t o yfi = 3 and d = 3 or 4, 30th cases lead t o T(M)= I2.31. a contradiction.

I

2


SHARPLY 2 - T R A N S I T I V E PERMUTATION GROUPS

939

PEOOP:It follows fioni ! I ] that xhe oniy solvable s u b g o u p in this class i.c Pfl'(4.3) x P R ( 3 , 3 ) in T = P S ( 1 2 . 3 ) . It is easy t o show thar T(M)= 12.9:.

Nest suppose that scc(G) = PR-(8! q ) and G contains a g ~ p aatorrxrpliism h

of order 3. In this case, all maximal subgroups o f @ are determined by K!eidnar: in : 7 : . Suppose that (G,.%f) is a cocntei. example t o Theorem 1.2: then by [ 7 ] ,

the mi: possibilities For M are


940

WANG

. . l i , .tschbac5er. G . \I. Seitz. Izrolurions in Shevalieg groups over fieids of '2'

s:iSgr:tc2s of exreptiom! groups zf Lie t y p e : p r e p ~ i n t . :j:1,H.

Con-gag, R. T.C;:xis, S.P.Xcrror,; R. A . Parker and R. -1, R'ilson,

.4tlns c,f,?r:te croups. Clarendor. Prtss. Oxford, 1385.

:7: P.B. Kleidman: The maxlxai sabgmups of t h Enit: 8-dirnessionai orthog-

onal g a u p s PQZ (g) and of their automnqh'!sm groups, J . .2lgsbra 1 P O f l P 8 7 ~ , 153-212. S ] P. 5 , fi1e;Jidman.The maxima; subgroups of the Steinberg trialicy groups

3D4!a) mc! of their aztnrnorphism groaps. J. .4/geQ1a 115 (l988!, 182-lG9. [?; P. B, Iileidnan, X i , IY. Liebec!:,

.Isunre? of the maxima! s l i h g r o ~ pof~ ~ Z P

k i t e simple g - r o q x : G'eorn. Dedicsre 2 5 (!3513). 37.5- 3800.

gronvs.

Lond. \lath. % c . L e c t . Note Series 1?P. Canbridge Vniversitp Press,

Carr,5r;cI$i:, !W9.

941


SHARPLY 2-TRANSITIVE PERMUTATION GROUPS

Received: Revised:

A p r i l 1991 July 1991