Sylow Subgroups and Nilpotent Subgroups ……………..........… Mashhour I.Mohammed
Sylow Subgroups and Nilpotent Subgroups
of GL(n,q)
Received: 5/3/2003
Accepted : 6/1/2004
Mashhour . I.Mohammed *
‫ملخص‬
Sylow r ‫الهدف من هذا البحث هو إعطاء الحد األعلى لرتبة الزمرة الجزئية من النوع‬
. GL(n,q) ‫لـ‬
Abstract
The aim of this paper is to give an upper bound for the order
of Sylow r -subgroups of GL(n,q).
0.Introduction:
In [3], Glauberman proved the following.
Theorem: Let V be a nontrivial finite dimensional GF(q)-vector space and let G
be a nilpotent subgroup of GL(V) that has order coprime to q and class at most
two. Then
G

V
.
In [2], Flavell proved the following.
Theorem: Let V be a n-dimensional GF (q)-vector space with n  1.
Let K  G  GL (V) and suppose that
nilpotent with class at most two. Then
(i)
G:K

V
G/K has order coprime to q and is
;
(ii) If f is a nondegenerate bilinear form on V that is preserved by G then
G:K
 ( n) [ n ]
 2  2
where  (n)  0 if q is even or n is even,
1 if q is odd and n is odd
Al-Manarah, Vol. 12, No. 2, 2006 .
161
Sylow Subgroups and Nilpotent Subgroups ……………..........… Mashhour I.Mohammed
*
Assistant Professor, Mutah University, Jordan .

8 if q = 3 or 5
= 1+q otherwise
--------------------------------------------------------------------------------1999 Mathematics Subject Classification 20 G 40
In this paper we determine an upper bound for the the order of Sylow rsubgroups of GL (n,q). First we have the following theorem:
Theorem 1: Let U be an irreducible subgroup of GL (n,q), where q =
p a ,p
is
prime and p | U . Let U = U1  U 2
for some groups U1 ,U 2 of U.Then U1 has
exactly one homogeneous component, that is , all irreducible U1 -submodules are
isomorphic.
1.Notation and Basic Definitions
Througout, all groups are finite.If H and X are subsets of a group G, then
C H (X) denotes the centralizer of X in H. If U and W are two subspaces of a
vector space V, we write U  W to denote their direct sum. If E  V is an
 Ex
x
irreducible R-submodule then E denotes the conjugation of E by x ,and xR
x
n
denotes < E : xR >. If n is an integer and p is a prime, then we denote by p
the p-part of n, i.e, the largest power of p that divides n . H  K will denote the
wreath product of H by K. For more information about wreath product see [1].
Z n denotes the cyclic group of order n and S n denotes the symmetric group of
degree n.
Lemma 1: Let U, U 1 and U 2 be as in Theorem 1,and let V be a nontrivial finite
dimensional GF(q) - vector space. Assume that E is an irreducible U 1 -submodule
 Ex
of V, then U leaves H = xU 2
Al-Manarah, Vol. 12, No. 2, 2006 .
x
invariant, where E is the conjugate of E.
162
Sylow Subgroups and Nilpotent Subgroups ……………..........… Mashhour I.Mohammed
E xg   E x  H
g 

U
x

U
xU 2
Proof: Let g 2 , then H = 2
.So U 2 leaves H invariant. If
z  U1 , then
Hz 
 E xz   E zx   E x  H
xU 2
xU 2
xU 2
(using the assumption that E is
an irreducible U 1 - submodule) . So U 1 leaves H invariant. Thus H is invariant
under U.
Lemma 2: Let U, U 1 , U 2 and V be as in Lemma 1. Then there exists an
m
irreducible U 1 -module such that V  E as a U 1 - module.
x
Proof: Simply we can take a minimal set E1 , E2 ,..., Em of conjugates of E ,
 Ex
such that
xU 2
= E1  E2  ....  Em . For E and E
x
we either have E = E
x
or E
 E  {0}. Assume that E 1 = E, If there exists.
x
 E 1 , then E 1  E 2 = {0} and E 1 +E 2 =
 E 1  E 2 , then E 3  ( E 1 +E 2 ) = {0}, and
 Ex
x
E 2 {E : x U 2 } such that E 2
E 1  E 2 . If there is E 3 = E
x
hence E1  E2  E3  E1  E2  E3 . As
irreducible, this implies that
 Ex
V = xU 2
= E1  E2  ...........  Em = E
m
xU 2
is invariant under U, and U is
as a U 1 -module.
Proof of Theorem 1: The proof is an immediate consequence of Lemma 1 and
Lemma 2.
Lemma 3: Let U be an irreducible subgroup of GL( n, q ) and let q be a power of
p such that p U . If U 1 and U 2 are two subgroups of U such that U = U 1  U 2 ,
b
then U 2  GL(m, q ) .
Proof : By Schur's Lemma, the centralizer
of
C
GF(q),
This
implies
that
is some extension field
U 2  CGL ( n, q ) (U1 ) = GL(m, q b )
(U 1 )  GF (q b ).
GL ( E )
Al-Manarah, Vol. 12, No. 2, 2006 .
CGL ( n, q ) (U1 )
163
where
Sylow Subgroups and Nilpotent Subgroups ……………..........… Mashhour I.Mohammed
Theorem 2: Let U be a nilpotent subgroup of GL (n,q). If r is a prime divisor
of
U
. Then
(i) U = U r  U r ,where U r is the Sylow r-subgroup of U and U r is the direct
product of all other Sylow t - subgroups of U, (t , r) = 1.
'
'
(ii) U r  GL(d,q) , where d = dim E and E is a faithful irreducible U r module.
Ur
'

b
GL(m,q ) if C GL ( E )
(U r )
b
= GF(q ).
r qb 1
(iii)
Proof:
(i) Since any nilpotent group is the direct product of its Sylow psubgroups, then (i) is clear.
(ii) : By Lemma 1,there exists a faithful irreducible U r -module E
such that V = E
 g1





as a U r - module. If g  U r then g can be written as





g1  m-blocks ,where g 1 = g | E , is the restriction of g on E
g1
So U r
m

GL (d,q).
(iii) : By Lemma 3,(ii) follows.


(iv) : Let R be an irreducible r-subgroup of GL(d,q). Since
C
GL ( d , q ) ( R)
b
= GF(q ) , then we may consider V =V(d,q) as a
d
vector space over the centralizer i.e V = V( q
Al-Manarah, Vol. 12, No. 2, 2006 .
164
, qb )

,R

d
GL( q
, qb )
. As
Sylow Subgroups and Nilpotent Subgroups ……………..........… Mashhour I.Mohammed

 Z( R )  C GL(V ) ( R) , it follows that

1

r
C
GL (V )
( R)
b
= q -1.
Theorem 3 : If r is a prime divisor of q-1, and r is odd or r = 2 and q

1 (mod 4), then the Sylow r-subgroups of GL (n,q) are isomorphic to Z ( q 1) r
H
the wreath product of Z ( q 1) r by H, H is a Sylow r- Subgroup of S n .
To prove Theorem 3, the following Lemma is of great importance .
Lemma 4 : If r is a prime divisor of q-1 , then the following hold.
(i) If r = 2 and q

i
3 (mod 4) , then (q 1) 2 = 2 if i is odd
i 2 (q  1) 2 if i is even
i
(ii) Otherwise (q -1) r = i r (q-1) r .
Proof : We first observe that if (s,r) = 1,then
1  q  ...  q s 1  1  1  ...  1  s  0(mod r ). Therefore
(1  q  ...  q s 1 ) r  1  s r .
s
Thus ( s, r )  1  (q  1) r  (q  1) r  (q  1) r s r ..
a
a
Let (q  1) r  r with a  1. Then q  1  r x, with
(1)
( x, r )  1, and
r
r 
q r  1  (1  r a x) r    r ja x j
j 1  j 
.
  r  ja j 
  r x  r  (rr a ) r  r a 1 .
 j


If j = 1, then   
  r  ja j 
  r x   (rr ja ) r  r ja1  r a 1 .
 j

2

j

r

1
,
r
If
then   
  r  ja j 
  r x   (r a r ) r  r ra  r 2 a  r a  a  r a 1
 j

 
r
, with equality if and
If j  r , then
only if r  2 and a  1 .
Al-Manarah, Vol. 12, No. 2, 2006 .
165
Sylow Subgroups and Nilpotent Subgroups ……………..........… Mashhour I.Mohammed
Therefore
(r , a)  (2,1)  (q r  1) r  r a 1  rr a  rr (q  1) r
(2)
Note also that ( r , a )  ( 2,1) if and only if r  2 and q  3(mod 4) .
We now prove (ii) by induction on i .Supposing that (i) holds for i
s
 s , we consider (q  1) r . If ( s, r )  1 Then our statement follows from
(1). Otherwise , s  rt and
(q s  1) r  (( q r ) t  1) r
r
= t r (q  1) r by the inductive assumption because t  s
= t r rr (q  1) r , by (2)
=
(tr) r q  1r  s r q  1r
, as desired.
Next, we prove (i). If i is odd, then (ii) follows from (1). Otherwise,


i  2 j and (q i  1) 2  q 2   1 2 ,
j
j 2 (q 2  1) 2
2
by (i) and because q  1(mod 4)
= j2 (q  1) 2 (q  1) 2  j2 2 2 (q  1) 2  (2 j ) 2 (q  1) 2
= i2 (q  1) 2 , as desired.
Proof of Theorem 3: Consider |GL(n,q)| =
n
GL(n,q) . So by Lemma 4, |GL(n,q)| r =

q
n
2
 
n
i 1

 qi 1 r
i 1


 qi 1 r
, the order of
=
n
n
((q-1) ) r
 (ir )
i 1
n
= ((q-1) ) r (n!). This implies that the Sylow r- subgroups T of
GL(n,q) are monomial i.e
T

Z(
q1 )r
of S n .
Al-Manarah, Vol. 12, No. 2, 2006 .
166
 H, where H is a Sylow r- subgroup
Sylow Subgroups and Nilpotent Subgroups ……………..........… Mashhour I.Mohammed
Theorem 4 : If r is an odd prime divisor of q-1, then |GL(n,q)| r
Proof: Consider the p- adic expression of n
 qn
p 1
i
 ni p
n = i 0
, the n i ,s are
i.e
p 1
uniquely determined , then (n!) r  r
( n
 ni ) /(r 1)
i 0

r ( n 1 ) / r 1 .
1 / r 1
 2 , we have
As r
(( q  1 )n )r .r n 1 / r 1  ( q  1 )r .2 n 1  ( q  1 )r ( 2( q  1 )r )n 1 If q-1 is not a power of r,
then ( q  1 )r  ( q  1 ) / 2 .
n
n 1 / r 1
 ( q  1 )r .( 2( q  1 )r )n 1
Hence (( q  1 ) )r .r
 ( q  1 )r ( q  1 )n 1  ( q  1 )n  q n .This proves the theorem.
Example: Consider the group GL(3,7). To find the Sylow 3-subgroups of
 1
{
2


GL(3,7),take D = 
i , i = 1 ,2 ,3, is a power of

 ,   GF ( 7 ) }
 i
3
 3 
, D = 3 and order .
r = 3 . Let H be a Sylow 3- subgroup of S 3 and
represent it by 3  3 permutation matrix and find H

such that H

0 1 0 
0 0 1





1 0 0 
H  GL(3,7). So D H = < D , 
> is a Sylow 3-subgroup of
4
3
GL(3,7) and |GL(3,7)| 3 = 3  7 .
Acknowledgment: My thanks are due to the referees for their remarks.
Al-Manarah, Vol. 12, No. 2, 2006 .
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16.25561
Sylow Subgroups and Nilpotent Subgroups ……………..........… Mashhour I.Mohammed
References
[ 1] M. Aschbacher ,Finite group theory,Cambridge University Press, Cambridge
1986.
[2] P. Flavell, Class two sections of finite classical groups, J. London Math. Soc .
(2) 52 (1995) 111-120.
[3] G. Glauberman, On Burnside’s other
.55(1975) 469-476.
Al-Manarah, Vol. 12, No. 2, 2006 .
168
p  q
theorem, Pacific J .Math