Strongly Closed Abelian 2-Subgroups of Finite Groups

55

STRONGLY CLOSED ABELIAN 2-SUBGROUPS OF FINITE GROUPS D a v i d M. G o l d s c h m i d t

R e c a l l t h a t for g r o u p s A

_ c l o_ s e_ d -i n ag

E

T

with

respect

5

G if

T

C a

E

G , we s a y t h a t A

g

A,

F.

and

G

ag

strongly E

T implies

A.

For t h e p a s t s e v e r a l y e a r s , I h a v e b e e n s t u d y i n g v a r i o u s s p e c i a l c a s e s of t h e a b o v e s i t u a t i o n , w i t h T abelian.

S y l ( G ) and A 2 These c o n d i t i o n s a r i s e whenever a 2 - l o c a l s u b g r o u p N (11) t

G

c o n t r o l s t h e f u s i o n of i t s 2 - e l e m e n t s ,

for i t i s e a s y t o s e e t h a t

i n t h i s c a s e , Z ( H ) i s s t r o n g l y c l o s e d i n a n y Sylow 2 - s u b g r o u p containing it.

C o n v e r s e l y , G l a u b e r m a n h a s shown t h a t if A i s

a b e l i a n a n d s t r o n g l y c l o s e d i n a Sylow 2 - s u b g r o u p

of G , t h e n NG(A)

c o n t r o l s t h e f u s i o n of i t s 2 - e l e m e n t s . T h e r e seems t o b e g o o d r e a s o n t o b e l i e v e t h a t i f A is. a b e l i a n a n d s t r o n g l y c l o s e d i n a Sylow 2 - s u b g r o u p

of G , t h e n t h e normal

c l o s u r e of A i n G h a s c o m p o s i t i o n f a c t o r s of known t y p e .

I hope t o

h a v e a p r o o f of t h i s r e s u l t s h o r t l y . N o t e t h a t any of t h e f o l l o w i n g h y p o t h e s e s i m p l i e s t h e e x i s t e n c e of' a s t r o r i g l y c l o s e d a b e l i a n 2 - s u b g r o u p :

a)

Sylow 2-subgroups

of G a r e a b e l i a n .

b)

Sylow 2 - subgroups

o f G a r e of t y p e Sz(q) or U 3 ( q ) .

c)

A Sylow 2 - s u b g r o u p

d)

G h a s a w e a k l y embedded 2 - l o c a l

e)

The weak c l o s u r e of a c e n t r a l i n v o l u t i o n i n i t s c e n t r a l i z e r is aoelian.

contains an i s o l a t e d involution. subgroup.

CGLDS C I I M I DT

56

S i m p l e g r o u p s j a t l o f ' y i n ~t h e a b o v e h y p o t h e s e s h a v e been

c l a s s i f i e d by v a r i o u s a u t h o r s .

Thus, i n a d d i t i o n t o p r o v i d i n g

n e w 1nforrnnt1on, t h e proposed. c l a s s i f i c a t i o n g e n e r a l i z e s t h e above r e s u l t s , and s h o u l d g i v e i n d e p e n d e n t p r o o f s f o r a ) , b), d ) and p o s s i b l y c ) .

Univttrsl t y o f C a l i f o r n i a , i i e r k e l e y