2-Signalizers of finite groups

2-SIGNALIZERS V.

D.

OF

FINITE

GROUPS

Mazurov

UDC 549.44

T h e s u b g r o u p H of f i n i t e g r o u p G is c a l l e d a 2 - s i g n a l i z e r of g r o u p G if t h e o r d e r [ H I and i n d e x IG':xv'g(~)l a r e odd [1]. T h o m p s o n [I] put f o r t h t h e h y p o t h e s i s t h a t 2 - s i g n a l i z e r s of f i n i t e s i m p l e g r o u p s are commutative. T h e p u r p o s e of t h i s n o t e i s to p r o v i d e a c o n t r a d i c t o r y e x a m p l e : In t h e s i m p l e g r o u p P S L 3 (q), w h e r e q - nilpotent 2-signalizer.

Proof.

In g r o u p G = S L 3 ( q ) , ( q ) - -

1 (mod 4), q ¢ 2 a - 1 (a i s an i n t e g e r ) t h e r e e x i s t s a n o n -

1 (rood 4), t h e s e t

~q =

Co°) 1

}

is a s u b -

g r o u p of o r d e r q2; t h e s e t

°~o~ i s a s u b g r o u p i s o m o r p h i c to G L 2 (q). C l e a r l y , B n o r m a l i z e s A. A c o m p a r i s o n of t h e o r d e r s of SL 3 (q) and G L 2 (q) s h o w s that B c o n t a i n s a Sylow 2 - s u b g r o u p of g r o u p G. In s u b g r o u p B the s e t

o

( 2÷/~)-~

i s a c o m m u t a t i v e s u b g r o u p of o r d e r q 2 _ 1. L e t C l - 2 b e the c o m p l e m e n t of C. T h e e l e m e n t ~ =

-01 _

i s c o n t a i n e d in B - C a n d n o r m a l i z e s C a n d , c o n s e q u e n t l y , C I. T h e r e f o r e -~= < C, ~ > is a s u b g r o u p of o r d e r 2 ( q 2 1). S i n c e q = - 1 (mod 4), D c o n t a i n s a Sylow 2 - s u b g r o u p of g r o u p B, i . e . , a Sylow 2 - s u b g r o u p of G. T h u s , C~ i s a 2 - s i g n a l i z e r of g r o u p G. If, in a d d i t i o n , q ~ 2 a - 1, w h e r e a i s an i n t e g e r , t h e n ( I CI [, q + 1) # 1 and, s i n c e (q + 1, I Z (G) I ) = 1, C 1 d o e s not b e l o n g to the c e n t e r Z(G) of g r o u p G. T h e c e n t r a l i z e r of s u b g r o u p A in g r o u p B is & n z(~; : t h e r e f o r e Cf d o e s not c e n t r a l i z e A w h e r e A i s a H a l l s u b g r o u p of g r o u p A C t ; t h e r e f o r e AC 1 is n o n - n i l p o t e n t . C l e a r l y , A C 1 (mod Z(G)) i s a n o n - n i l p o t e n t 2 - s i g n a l i z e r of g r o u p G/ZCG) - PSL~Cq). W e s h a l l a l s o show t h a t t h e r e e x i s t s i m p l e g r o u p s w i t h n i l p o t e n t 2 - s i g n a l i z e r s f o r a n y p r e v i o u s l y specified nilpotent class. If q i s the i n t e g r a l p o s i t i v e e x p o n e n t of odd p r i m e p and r~ = ~k+~ _ ¢, w h e r e k i s a p o s i t i v e i n t e g e r , t h e n t h e r e e x i s t s a n i l p o t e n t 2 - s i g n a l i z e r of c l a s s k in PSL ~ (9)" Proof.

L e t G=SL,., (c/) w h e r e ,~

and q a r e d e f i n e d a b o v e .

L e t A b e the s e t of a l l e l e m e n t s of g r o u p

T r a n s l a t e d f r o m A l g e b r a i L o g i k a , Vol. 7, No. 3, pp. 6 0 - 6 2 , M a y - J u n e , 1968. O r i g i n a l a r t i c l e s u b m i t t e d M a y 14, 1968.

167

G of f o r m

X,o E, .

.

.

0

.

•

.

•

°

.

.XK~,,XK2 ,..'., E,~ w h e r e E i i s a unit m a t r i x of d i m e n s i o n a l i t y 2 i x 2 i, Xij i s a m a t r i x of d i m e n s i o n a l i t y 2 i × 2J w i t h e l e m e n t s of t h e f i e l d G F (q). C l e a r l y , A - p is a s u b g r o u p of g r o u p G and the c l a s s of n i l p o t e n c y A is e q u a l to k. L e t B b e a s u b g r o u p of g r o u p G c o n s i s t i n g of a l l e l e m e n t s of f o r m

o)

k

"4~

0 where

,'~iEdL2L ( ~ )

• B

normalizes subgroup A and t3n~q=.

Since

IGI = (q'~- ¢ ) ( q n - ~ ) ' " C ~ ' ~ - ~ - ' )

k

,! "-

Zi

I~1 = / - 7 c~ - , ) ( ~ - ~ . . .

Zi

cg - g

,

Zi - f )

,

d i r e c t v e r i f i c a t i o n s h o w s t h a t I G : B I i s an odd n u m b e r . T h u s , A is a 2 - s i g n a l i z e r of g r o u p G and, in o r d e r to p r o v e o u r s t a t e m e n t , it i s s u f f i c i e n t to note t h a t t h e i n t e r s e c t i o n of s u b g r o u p A w i t h the c e n t e r of g r o u p G is t r i v i a l . LITERATURE 1.

168

CITED

J . G . T h o m p s o n , " 2 - s i g n a l i z e r s of f i n i t e g r o u p s , " P a c i f . J . M a t h . , 14, No. 1, 3 6 3 - 3 6 4 (1964).