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Flag-Transitive Subgroups of Chevalley Groups
122
FLAG-TRANSITIVE SUBGROUPS OF CHEVALLEY GROUPS Gary M. Seitz
Let G be a finite group of Chevalley type and B the Bore1 If L
subgroup of G.
5 G and G
BL, then L is said to be flag-
=
In [ 3 ] D. Higman determined all flag-transitive sub-
transitive.
g r o u p s of G in case G is of type
A,.
The main result here is an
extension of Higman's theorem to the case of a general group of The result is then used to show the nonexistence
Chevalley type.
of 2-transitive permutation representations of certain groups of Chevalley type. Let G
0
be a Chevalley group of normal or twisted type such Let Go
that Z(Go) = 1.
2 G1 5 Aut(Go) where GI
and the diagonal and field automorphisms of Go.
is generated b y G
0 We say that G is of
. If G is of Chevalley type and 1 B the Bocel subgroup of G , let U = Fit(B). Chevalley type provided Go
5 G 5 G
G t]e 2 group of Chevalley
THEOREM 1. suppose L 5 G
2
of rank n
Then either UG
flag-transitive.
2
2
and
5 L or one of the
following holds:
1)
G
g
PSL(3,2)
and
ILI
3'7.
=
2
ii)
G
PTL(3,8) and ILI= 3 -73.
iii)
G
PSL(4,2)
iv)
G
v)
G
Vi)
G
and L = G I . 2 F4(2) and L = G'.
vii)
G
PSp(4,3)
g
Sp(4,2) a
and L 8S 6 g@ L
A
A
A
7
.
6'
G2(2)
PSU(4,2)
and L 2 2
maximal parabolic
subgroup of PSU(4,2) of order 2 6 * 3 . 5 .
FLAG TRANSITIVE GROUPS
123
If G is a group o f Chevalley type, we write G = G(q) to indicate that G is defined over the field P
9
.
As application o f
Theorem 1 we have the following results.
..., sn
,n 2 2 be an indecomposable coxeter group. There is an integer N, depending 0” W, so that if G = G(q)
THEOREM 2.
is a -
__ Let W =
sly
group of Chevalley e-
Sype W,
then
with q
G has no faithful 2-transitive permutation representa-
A n-and G & represented on the cosets of
tion unless _ _ W is of ~type
maximal parabolic subgroup THEOREM 3.
2 N and having Weyl group o f
Let
G
of
a
G.
a group o f Chevalley type with UG
q > 2, P S U ( ~ , ~ ) ,P S U ( ~ , ~ ) , Gz(q), q > 2,
or
3
D4(q).
PSp(Q,q),
Then G has no
non-trivial 2-transitive permutation representation. The connection between Theorems 2 and 3 and Theorem 1 can easily be seen as follows. type with q = p
Let G = G(q) be a group o f Chevalley
a
and p prime. Suppose that G is 2-transitive on the G cosets of L 5 G. Then lL = 1 + 0 for (3 an irreducible character of
G.
If ple(1) then 1G:LI is prime to p and L contains a Sylow
p-subgroup of G.
In this case it easily follows that L is a maximal
parabolic subgroup and consequently G is of type An. Suppose G + xr when x l,...yxr are irrep ,/’ e(1). Write lg = 1 + x1 +
...
ducible characters o f G. i = 1,...,r then 1 = ( 1 G.
G
If we know that p I x (1) for each i G , lg ) = the number of L,B-double cosets of
Thus G = BL and L is flag-transitive.
question is whether or not p
I
Thus the only remaining
xi(l) for each i = 1, ..., r.
applying a result of Green [ Z ] we obtain Theorem 2.
By
Theorem 3
follows from work of Curtis, Iwahori, and Kilmoyer [l]. From the above it is clear that Theorem 1 can be used to
determine all 2-transitive permutation representations of all the
124
SEITZ
g r o u p s of C h e v a l l e y t y p e once i t i s known t h a t w i t h o n l y a few e x c e p t i o n s p d i v i d e s t h e d e g r e e of a l l n o n - p r i n c i p a l G
c o n s t i t u e n t s o f lg
irreducible
f o r G a g r o u p of C h e v a l l e y t y p e and o f
characteristic p. The p r o o f o f Theorem I p r o c e e d s as f o l l o w s . c h a r a c t e r i s t i c of G , show t h a t
The b a s i c aim i s t o
s o t h a t U is a p-group.
a l a r g e p o r t i o n of
U i s contained. i n
show t h a t w i t h few e x c e p t i o n s L of D i n P , where P i s
Let p be the
L, I n p a r t i c u l a r w e
P is 2-transitive
a p a r a b o l i c s u b g r o u p of
on t h e c o s e t s
G g e n e r a t e d by R and
one o f t h e f u n d a m e n t a l r e f l e c t i o n s g e n e r a t i n g t h e W e y l g r o u p o f G . Once t h i s i s done for e a c h s u c h P , t h e p r o o f i s c o m p l e t e d r a t h e r easily.
I t i s e a s y t o see t h a t we n e e d o n l y c o n s i d e r t h e c a s e where
IG:U
G
I
i s prime t o p .
bility that L
The f i r s t t r o u b l e s o m e p o i n t i s t h e p o s s i -
U = 1 and L is a p ' - g r o u p .
Here t h e i d e a i s t o
f i n d a low d i m e n s i o n a l ( p r o j e c t i v e ) r e p r e s e n t a t i o n of UG and t h e n l i f t t h i s t o a complex r e p r e s e n t a t i o n of L .
Then f i n d l n g l a r g e
prime d i v i s o r s o f ILI we a r e a b l e t o a p p l y r e s u l t s o f F e i t t o o b t a i n i n f o r m a t i o n a b o u t t h e s t r u c t u r e o f L , and t h j s l e a d s t o a contradiction. q = 2.
Next w e s e p a r a t e l y h a n d l e t h e c a s e G
= i;(q) and
U s i n g a n i n d u c t i v e l e m m a r e g a r d i n g t h e r a n k o f G we c a n
r e d u c e t o c a s e o f a g r o u p of r a n k a t most 4 , w i t h t h e e x c e p t i o n o f Sp(2n,2). with n
A f t e r g i v i n g a s e p a r a t e argument t o e l i m i n a t e S p ( 2 n , 2 )
2 3 w e u s e o r d e r c o n s i d e r a t i o n s for t h e r e m a i n i n g c a s e s .
t h r o u g h o u t t h i s s e c t i o n of t h e p a p e r and l a t e r o n e s d e t a i l e d i n f o r m a t i o n c o n c e r n i n g t h e s t r u c t u r e o f U i s u s e d when G i s o f r a n k 2 . Next w e assume q > 2 , l e a s t one 2 - t r a n s i t i v e
L
U # 1, a n d show t h a t t h e r e i s a t
s e c t i o n as d e s c r i b e d a b o v e .
Once t h i s i s
a c c o m p l i s h e d we u s e t h e f a c t t h a t t h e Dynkin d i a g r a m i s c o n n e c t e d ,
FLAG TRANSITIVE GROUPS
125
r e d u c e t o t h e c a s e o f a r a n k 2 g r o u p , and show a l l t h e a p p r o p r i a t e s e c t i o n s are 2 - t r a n s i t i v e .
REFERENCES
113
C . W.
Curtis, N.
I w a h o r i , a n d R. K i l m o y e r , Hecke
a l g e b r a s and c h a r a c t e r s o f p a r a b o l i c t y p e o f f i n i t e groups with (B,N)-pairs, [2]
J. A.
Math. P u b . I . H . E . S . ,
G r e e n , On t h e S t e i n b e r g c h a r a c t e r o f f i n i t e
C h e v a l l e y g r o u p s , Math. Z e i t
[3]
( t o appear).
. 117 ( 1 9 7 0 ) , 272-288.
D . G . Higman, F l a g - t r a n s i t i v e
c o l l i n e a t i o n g r o u p s of
f i n i t e p r o j e c t i v e s p a c e s , I l l . J . Math.
U n i v e r s i t y o f Oregon
6
( 1 9 6 2 ) , 434-446.
FLAG-TRANSITIVE SUBGROUPS OF CHEVALLEY GROUPS Gary M. Seitz
Let G be a finite group of Chevalley type and B the Bore1 If L
subgroup of G.
5 G and G
BL, then L is said to be flag-
=
In [ 3 ] D. Higman determined all flag-transitive sub-
transitive.
g r o u p s of G in case G is of type
A,.
The main result here is an
extension of Higman's theorem to the case of a general group of The result is then used to show the nonexistence
Chevalley type.
of 2-transitive permutation representations of certain groups of Chevalley type. Let G
0
be a Chevalley group of normal or twisted type such Let Go
that Z(Go) = 1.
2 G1 5 Aut(Go) where GI
and the diagonal and field automorphisms of Go.
is generated b y G
0 We say that G is of
. If G is of Chevalley type and 1 B the Bocel subgroup of G , let U = Fit(B). Chevalley type provided Go
5 G 5 G
G t]e 2 group of Chevalley
THEOREM 1. suppose L 5 G
2
of rank n
Then either UG
flag-transitive.
2
2
and
5 L or one of the
following holds:
1)
G
g
PSL(3,2)
and
ILI
3'7.
=
2
ii)
G
PTL(3,8) and ILI= 3 -73.
iii)
G
PSL(4,2)
iv)
G
v)
G
Vi)
G
and L = G I . 2 F4(2) and L = G'.
vii)
G
PSp(4,3)
g
Sp(4,2) a
and L 8S 6 g@ L
A
A
A
7
.
6'
G2(2)
PSU(4,2)
and L 2 2
maximal parabolic
subgroup of PSU(4,2) of order 2 6 * 3 . 5 .
FLAG TRANSITIVE GROUPS
123
If G is a group o f Chevalley type, we write G = G(q) to indicate that G is defined over the field P
9
.
As application o f
Theorem 1 we have the following results.
..., sn
,n 2 2 be an indecomposable coxeter group. There is an integer N, depending 0” W, so that if G = G(q)
THEOREM 2.
is a -
__ Let W =
sly
group of Chevalley e-
Sype W,
then
with q
G has no faithful 2-transitive permutation representa-
A n-and G & represented on the cosets of
tion unless _ _ W is of ~type
maximal parabolic subgroup THEOREM 3.
2 N and having Weyl group o f
Let
G
of
a
G.
a group o f Chevalley type with UG
q > 2, P S U ( ~ , ~ ) ,P S U ( ~ , ~ ) , Gz(q), q > 2,
or
3
D4(q).
PSp(Q,q),
Then G has no
non-trivial 2-transitive permutation representation. The connection between Theorems 2 and 3 and Theorem 1 can easily be seen as follows. type with q = p
Let G = G(q) be a group o f Chevalley
a
and p prime. Suppose that G is 2-transitive on the G cosets of L 5 G. Then lL = 1 + 0 for (3 an irreducible character of
G.
If ple(1) then 1G:LI is prime to p and L contains a Sylow
p-subgroup of G.
In this case it easily follows that L is a maximal
parabolic subgroup and consequently G is of type An. Suppose G + xr when x l,...yxr are irrep ,/’ e(1). Write lg = 1 + x1 +
...
ducible characters o f G. i = 1,...,r then 1 = ( 1 G.
G
If we know that p I x (1) for each i G , lg ) = the number of L,B-double cosets of
Thus G = BL and L is flag-transitive.
question is whether or not p
I
Thus the only remaining
xi(l) for each i = 1, ..., r.
applying a result of Green [ Z ] we obtain Theorem 2.
By
Theorem 3
follows from work of Curtis, Iwahori, and Kilmoyer [l]. From the above it is clear that Theorem 1 can be used to
determine all 2-transitive permutation representations of all the
124
SEITZ
g r o u p s of C h e v a l l e y t y p e once i t i s known t h a t w i t h o n l y a few e x c e p t i o n s p d i v i d e s t h e d e g r e e of a l l n o n - p r i n c i p a l G
c o n s t i t u e n t s o f lg
irreducible
f o r G a g r o u p of C h e v a l l e y t y p e and o f
characteristic p. The p r o o f o f Theorem I p r o c e e d s as f o l l o w s . c h a r a c t e r i s t i c of G , show t h a t
The b a s i c aim i s t o
s o t h a t U is a p-group.
a l a r g e p o r t i o n of
U i s contained. i n
show t h a t w i t h few e x c e p t i o n s L of D i n P , where P i s
Let p be the
L, I n p a r t i c u l a r w e
P is 2-transitive
a p a r a b o l i c s u b g r o u p of
on t h e c o s e t s
G g e n e r a t e d by R and
one o f t h e f u n d a m e n t a l r e f l e c t i o n s g e n e r a t i n g t h e W e y l g r o u p o f G . Once t h i s i s done for e a c h s u c h P , t h e p r o o f i s c o m p l e t e d r a t h e r easily.
I t i s e a s y t o see t h a t we n e e d o n l y c o n s i d e r t h e c a s e where
IG:U
G
I
i s prime t o p .
bility that L
The f i r s t t r o u b l e s o m e p o i n t i s t h e p o s s i -
U = 1 and L is a p ' - g r o u p .
Here t h e i d e a i s t o
f i n d a low d i m e n s i o n a l ( p r o j e c t i v e ) r e p r e s e n t a t i o n of UG and t h e n l i f t t h i s t o a complex r e p r e s e n t a t i o n of L .
Then f i n d l n g l a r g e
prime d i v i s o r s o f ILI we a r e a b l e t o a p p l y r e s u l t s o f F e i t t o o b t a i n i n f o r m a t i o n a b o u t t h e s t r u c t u r e o f L , and t h j s l e a d s t o a contradiction. q = 2.
Next w e s e p a r a t e l y h a n d l e t h e c a s e G
= i;(q) and
U s i n g a n i n d u c t i v e l e m m a r e g a r d i n g t h e r a n k o f G we c a n
r e d u c e t o c a s e o f a g r o u p of r a n k a t most 4 , w i t h t h e e x c e p t i o n o f Sp(2n,2). with n
A f t e r g i v i n g a s e p a r a t e argument t o e l i m i n a t e S p ( 2 n , 2 )
2 3 w e u s e o r d e r c o n s i d e r a t i o n s for t h e r e m a i n i n g c a s e s .
t h r o u g h o u t t h i s s e c t i o n of t h e p a p e r and l a t e r o n e s d e t a i l e d i n f o r m a t i o n c o n c e r n i n g t h e s t r u c t u r e o f U i s u s e d when G i s o f r a n k 2 . Next w e assume q > 2 , l e a s t one 2 - t r a n s i t i v e
L
U # 1, a n d show t h a t t h e r e i s a t
s e c t i o n as d e s c r i b e d a b o v e .
Once t h i s i s
a c c o m p l i s h e d we u s e t h e f a c t t h a t t h e Dynkin d i a g r a m i s c o n n e c t e d ,
FLAG TRANSITIVE GROUPS
125
r e d u c e t o t h e c a s e o f a r a n k 2 g r o u p , and show a l l t h e a p p r o p r i a t e s e c t i o n s are 2 - t r a n s i t i v e .
REFERENCES
113
C . W.
Curtis, N.
I w a h o r i , a n d R. K i l m o y e r , Hecke
a l g e b r a s and c h a r a c t e r s o f p a r a b o l i c t y p e o f f i n i t e groups with (B,N)-pairs, [2]
J. A.
Math. P u b . I . H . E . S . ,
G r e e n , On t h e S t e i n b e r g c h a r a c t e r o f f i n i t e
C h e v a l l e y g r o u p s , Math. Z e i t
[3]
( t o appear).
. 117 ( 1 9 7 0 ) , 272-288.
D . G . Higman, F l a g - t r a n s i t i v e
c o l l i n e a t i o n g r o u p s of
f i n i t e p r o j e c t i v e s p a c e s , I l l . J . Math.
U n i v e r s i t y o f Oregon
6
( 1 9 6 2 ) , 434-446.