THE JACOBSON RADICAL OF GROUP ALGEBRAS
NORTH-MOLIAND MATHEMATICS STUDIES Notas de Matematica ( 115)
Editor: Leopoldo Nachbin Centro Brasileiro de Pesquisas Fisicas Rio de Janeiro and University of Rochester
NORTH-HOLLAND -AMSTERDAM
NEW YORK
OXFORD TOKYO
135
THE JACOBSON RADICAL OF GROUP ALGEBRAS G regory KARPlLOVSKY Department of Mathematics Universityof the Witwatersrand Johannesburg, SouthAfrica
1987
NORTH-HOLLAND-AMSTERDAM
0
NEW YORK
OXFORD .TOKYO
@
ElsevierScience Publishers B.V., 1987
All rights reserved. No part of this publication may be reproduced, stored in a retrievals ystem, ortransmitted, in any form orby any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.
ISBN: 0 444 70190 7
Publishers:
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Library of Congress Cataloging-inPubliition Data
Karpilovsky, Gregory, 1940The Jacobson r a d i c a l of group algebras. (North-Holland mathematics s t u d i e s ; 135) (Notas de ; 115) mate&ica Bibliography : p Includes index. 1. Group algebras. 2. Jacobson r a d i c a l . 3. Modules (Algebra) I. T i t l e . 11. S e r i e s . 111. Series: Notas de m a t e d t i c s (Rio de Janelpo, Brazil) ; no. 115. QAl.N86 no. 115 CgA1711 510 s E512'.243 86-32918 ISBN 0-444-70190-7 (U.S. )
.
PRINTED IN THE NETHERLANDS
Ila the memory a€ my father
ILIA KARPICSUtiKq
(1915-1942)
This Page Intentionally Left Blank
vii
Preface
Let
G be a f i n i t e group and l e t F be a f i e l d .
r e p r e s e n t a t i o n s of algebra
FG.
radical
JIFG) of
G
over
F
I t i s w e l l known t h a t l i n e a r
can be i n t e r p r e t e d a s modules over t h e group
Thus t h e i n v e s t i g a t i o n of r i n g - t h e o r e t i c s t r u c t u r e of t h e Jacobson
FG
i s of fundamental importance.
During t h e l a s t two
decades t h e s u b j e c t has been pursued by a number of r e s e a r c h e r s and many i n t e r e s t i n g r e s u l t s have been obtained.
The p r e s e n t book i s intended t o g i v e a
systematic account of t h i s work b u t t h e treatment i s by no means intended t o be exhaustive.
The author has t r i e d t o be f a i r l y complete i n what he c o n s i d e r s a s
t h e main body of t h e theory, and t h e r e a d e r should g e t a considerable amount of knowledge of c e n t r a l i d e a s , t h e b a s i c r e s u l t s , and t h e fundamental methods. have t r i e d t o avoid making t h e d i s c u s s i o n t o o t e c h n i c a l .
We
With t h i s view i n mind,
maximum g e n e r a l i t y has not been achieved in those p l a c e s where t h i s would e n t a i l a l o s s of c l a r i t y o r a l o t of t e c h n i c a l i t i e s . The p r e s e n t monograph i s w r i t t e n on t h e assumption t h a t t h e r e a d e r has had t h e equivalent of a standard f i r s t - y e a r graduate a l g e b r a course.
Thus we assume a
f a m i l i a r i t y with b a s i c r i n g - t h e o r e t i c and group-theoretic concepts and an understanding of elementary p r o p e r t i e s of modules, t e n s o r products, and f i e l d s .
For
t h e convenience of t h e r e a d e r , a chapter on a l g e b r a i c p r e l i m i n a r i e s i s included. This chapter provides a survey of t o p i c s needed l a t e r i n t h e book.
There i s a
f a i r l y l a r g e bibliography of works which a r e e i t h e r d i r e c t l y r e l e v a n t t o t h e t e x t o r o f f e r supplementary m a t e r i a l of i n t e r e s t . A word about n o t a t i o n .
As is customary, Theorem 4.3.2 denotes t h e second
r e s u l t of Section 3 of Chapter 4 ;
however, f o r s i m p l i c i t y , a l l r e f e r e n c e s t o t h i s
viii
PREFACE
r e s u l t within Chapter 4 i t s e l f a r e designated a s Theorem 3 . 2 . A systematic d e s c r i p t i o n of t h e m a t e r i a l is supplied by t h e i n t r o d u c t i o n s t o
i n d i v i d u a l chapters and t h e r e f o r e w i l l not be repeated here. I would l i k e t o express my g r a t i t u d e t o my wife f o r t h e encouragement she has
given me i n t h e p r e p a r a t i o n of t h i s book.
My thanks t o P.Brockhaus,
E.Fomanek,
S.Koshitani, B.Kiilshammer, K.Motose, Y.Ninomiya, Y.Tshushima and D.A.R.Wallace
for
sending me r e p r i n t s and p r e p r i n t s of t h e i r work, and a l s o t o Lucy Rich f o r her e x c e l l e n t typing.
Gregory Karpilovsky
ix
Contents
vii
PREFACE CHAPTER 1. RING-THEORETIC BACKGROUND 1. Notation and terminology 2. Artinian and noetherian modules Completely reducible modules 3. 4. Direct decomposition of rings Matrix rings 5. 6. The radical and socle of modules and rings 7. The Krull-Schmidt theorem 8. Projective, injective and flat modules 9. Projective covers 10. Algebras over fields
CHAPTER 2.
CHAPTER 3 .
1 1 5 12 15
17 22
36 40
46 52
GROUP ALGEBRAS AND THEIR MODULES
67
1. Group algebras 2. Central idempotents 3. The number of irreducible FG-modules 4. The induced modules 5. Relative projective and injective modules 6. Vertices of FG-modules
67 73 79 82 91 100
THE JACOBSON RADICAL OF GROUP ALGEBRAS: FOUNDATIONS OF THE THEORY 1. 2. 3. 4. 5.
6. 7. 8. 9. 10. 11. 12.
Elementary properties Direct products A characterization of elements of J(FG): the general case Conlon's theorem, Fong's dimension formula and related results A characterization of elements of J(FG); G is p-solvable A characterization of elements of J(Z(FG)) Frobenius groups Upper and lower bounds for A characterization of dimFJ(Z( dim# G ) ) and its applications Morita's theorem ~n application: criteria for J V G ) = FG*J(Z(FG)) Group algebras with radicals of square zero
105
105 116 122 140
171 177 183 191 204 228 243 248
CONTENTS
X
13. 14.
15. 16. 17.
CHAPTER 4:
Group algebras with central radicals Commutativity of the radical of the principal block Criteria for the commutativity of J ( F G ) The radical of blocks and normal subgroups Group algebras with radicals expressible as principal ideals
GROUP ALGEBRAS OF p-GROUPS OVER FIELDS OF CHARACTERISTIC
p
1. Dimension subgroups in characteristic p > 0 and related results 2. Computation of t ( P ) for some individual p-groups P characterization of groups P of order pa with 3. t(P) 4.
CHAPTER 5:
CHAPTER 6 :
CHAPTER 7:
a(p-1) + 1 ,
t ( P ) = pa, t ( P )
=
301
301 315
p'-l+p-l,
t ( P ) = pa-' and t ( P ) = (a+l)(p-1) + 1 Characterizations of p-groups P with t ( P )
322 328
...>v
is a composition series of minimal length for V, V1
after a finitely many terms.
has a composition series.
Conversely, suppose that V
then
V
9
then
V/Vl
is irreducible and
The desired conclusion is there-
Assume that
v = v0 3 v1 3 . . . 3 v n = o and
v = w0 are two composition series of case n
V.
3 w 1
>...>w m
Then these two series are called eqUiVaZent in
and there is a permutation cr
= rn
2.11. PROPOSITION.
= o
of
(The Jordan-HElder Theorem).
1,2,. .
.,721
such that
composition series, then every pair of composition series for We use induction on c ( V ) ,
Proof.
composition series for c(V) =
V.
where c ( V )
v
has a are equivalent.
is the minimum length of
The case c ( V ) = 1 being trivial, we may assume that
> 1 and that any module with a composition series of smaller length has
YZ
all of its composition series equivalent.
v = vo
U)
v
If a module
2
v
2
Fix a composition series
... 3 vn = 0
of minimal length and choose any other composition series for
v
(2)
If V
1
= =
w0
2
w1
2
... 3 wrn=
0
W , then by the induction hypothesis and in view of c ( V 1
) 4
n-1,
(1)
CHAPTER L
10
and (2) are equivalent. V +W 1
= 1
V
V/Vl
(3)
whence
Vl
since V Wl/(Vl
n
W1)
and
V/W
1
1
V /(V 1
w1
n W1 is maximal in both
W
f
1
V.
is a maximal submodule of
1
2
V
Hence we may assume that
1
in which case
It follows that
I n W 1 )
V
and
1
.
By Proposition 2.10, wl
has a composition series
v 1 n w 1 = x0 3 x 1 3 . . . 3 x k = o and therefore V1 3 Xo 3
... 3 Xk
are composition series for composition series for
V1
V
1
=
0
and
W
and W1.
2
Xo
2
... 2 X k
e(v
Because
)
< n,
=
0
every two
are equivalent, so the two series
v = v0 3 v1 3 v 2 > . . . 3 v n = o and
v = v o 3 v1 are equivalent.
2 x
0
3 . . . > xk = o
It follows that k < n - 1
two composition series for
W1
and e(W ) < n , proving that every 1
are equivalent.
The conclusion is that the two
series
v = w o 2 w1
2...2w
m
= o
and
v = w0 > w 1 are equivalent.
Since, by ( 3 ) ,
3 x
V/Vl
0
3...3xk = o
W1/Xo
2
and
V/Wl
2
Vl/Xo,
the series (1)
and ( 2 ) are also equivalent, a s required. A
module
Zength.
v
that is both artinian and noetherian is said to be of f i n i t e
In view of the Jordan-Hglder Theorem, for such a module
define its (composition) Zength
c(V)
unambiguously by
V
we can
ARTINAIN AND NOETHERIAN MODULES
c(V)
=
I
0
if V = O
n
if
has a composition series of length n.
V
... @ Vn
Let R R = V 8
2.12. PROPOSITION.
11
with V . 2
f
0
for i = 1,2
,..., n,
and write 1
...
e +
=
1
Then
V.
=
+
(ei E V . )
en
2
{e ,...,e is a set of pairwise orthogonal idempotents in R and n Re 1 Q i G n. Conversely, if { e ,. ..,e 1 is a set of pairwise orthogonal i'
idempotents in R, then
n
R(
1 ei) =
i=1
Given r E R, we have r
Proof.
=
n 8 Rei i= 1
r-1 =
1-P =
re +
... +
re,.
Thus in
particular
ei
=
+ e.e 2 1
... + e.e z n
for all i , j . Moreover Re. c V . and 22 z i = Gijei @ Ren which implies that Rei = Vi for all i
proving that e .e
R R = Re 8 1
...
.
Conversely, assume that { e
n potents and put e
n
Re
1 ei.
is a set of pairwise orthogonal idemand eei = eie = ei' for all i. Thus
Then e2 = e
i=1
1 Rei.
=
=
,...,en 1
If
Iriei
= 0
then multiplication on the right by
that r.e = 0 for all j . J j 2.13. COROLLARY. a set {e n
1=
'
,...,e n 1
I ei.
ej implies
n
i=1
Hence Re = 8 Rei, as required. i=1
Assume that R
is artinian or noetherian.
Then there exists
of pairwise orthogonal primitive idempotents of R
with
Furthermore
i=1
RR where each Re Proof.
=
Re 8
... 8 Ren
i is an indecomposable module.
Direct consequence of Propositions 2.12 and 2.9.
We shall refer to the R-modules Re ],...,Ren
in Corollary 2.13 as being the
CHAPTER I
.^
IL
prfncipal MecanposabZe R-modules. 3 . COMPLETELY REDUCIBLE MODULES. An
V
R-module
is said to be completely reducible if every submodule of
direct sunnnand of 3.1. PROPOSITION.
(il
Let V be completely reducible.
every submodule of V
module W '
(i) If W of
Let W
(ii)
verify that U module of thus U
W
module
W 8 K'
=
V
0.
V.
V
V
and
V.
is completely reducible.
contains an irreducible submodule.
Then W
1
V/W'
be a submodule of
V
with
By (i), it suffices to
So assume that
V = V @ V 1
.
2
for some sub-
as required.
and let U = V / K .
is completely reducible. Choose V 2
= W'
V/W
and
then V = K 8 K'
V,
is a submodule of
Then
u
=
U1 =
vl/W
is a sub-
u
(v +
w)/W and
1
8
is completely reducible.
(iii) Let
v=
is isomorphic to a submodule of
is isomorphic to a homomorphic image of
V # 0 then
Proof.
W'
V
Every submodule and every homomorphic image of
(iii) If
is a
V.
Every homomorphic image of
(ii)
V
2,
of
be a nonzero element of V
1
of
v.
for some nonzero submodules
If K '
W ,W 1
it follows that U 4 W 8 Wi maximality of
By Zorn's lemma, there exists a sub-
maximal with respect to the property that
for some submodule kr'
K 8 W2
V.
W.
9
w.
Write
is not irreducible, then
of
2
for either i = 1 or
2,
i
V.
= 2,
Because
contrary to the
Hence W' is irreducible.
The following alternative description of completely reducible modules is often useful. 3 . 2 . PROPOSITION.
For a nonzero module V
the following statements are equiva-
lent: (il
V
is completely reducible
(ii1
V
is the direct sum of irreducible submodules
13
c~PLETELYREDUCIBLE MODULES
(iii) V
is the sum of irreducible submodules. (i) * (ii): Consider the collection of sets of irreducible submodules
Proof. of
V
whose sum is direct.
By Proposition 3.1(ii), it is nonempty and, by
{Vi>, in this collection.
Zorn's lemma, there is a maximal element, say
W
=
W'
8 V
i and let V
W 8 W'.
=
If
W'
0 then by Proposition 3.1 (ii),(iii),
#
Thus W
contains an irreducible submodule V ' .
{Vi}.
to the maximality of
Hence
W'
Let
=
+
V ' = V ' 8 (8 Vi),
V = W,
0 and
contrary
as asserted.
(ii) * (iii): Obvious (iii)
Let W be a submodule of
(i):
a submodule W'
W+W'
of contradiction that W 8 W' # V thesis, V of
=
V
for i =
V
V . n ( W O W'I 3
W 8 W'
We are therefore left to verify that
W8W'.
=
Owing to Zorn's lemma, there exists
maximal with the property that W n W'
V
of
V.
... + Vn 1,2,...,n.
+
+ vi -
V . n ( W 8 W') = 0. 3
where
Since
and choose
vi
E Vi
w'
E V
Vi
with
V
=
0.
Hence
v.
9 W 8
Assume by way
w'.
By hypo-
is an irreducible submodule
for some j and so
is irreducible, we conclude that
Thus
W O W' + V . 3
and so W n (W' 8 V . ) 3 3.3. COROLLARY.
and
Hence V . 4 W @ 3
Vj
2,
=
=
Let
= WOW
' 8 vj
0, contrary to the maximality of W'. V # 0 be completely reducible.
.
The following are equi-
valent (i)
V
is artinian
(ii) V
is noetherian
(iii) V
is the direct sum of a finite number of irreducible submodules.
Proof. Direct consequence of Corollary 2.8 and Proposition 3.2. 3.4. PROPOSITION.
.
If R R is completely reducible, then every R-module is
completely reducible. proof. which sends
Let
V
p
to
Proposition 3.1(ii).
be an R-module and let v E V . M.,
is a homomorphism. Since V
5 1 Ro, 6 V
Thus
Then the map from R R to
RV
Rv is completely reducible by
the result follows by virtue of
CHAPTER I
14
Proposition 3.2. As
a preparation for the proof of the next result, we record the following
observation. Let V
3 . 5 . LEMMA.
=
8 Vi where each Vi
irreducible submodule of
V.
Then W
Proof. Choose nonzero w E W then the map sending rW =
RIJ
where j ranges over those i
Vi = Rvi.
to
into
PV
and write w for P E R
i
Since W,Vi
= CV
V
Vi
E Vi.
If
W
2
Vi
# 0
as required.
is said to be homogeneous if it can be
V
The sum of all irreducible
which are isomorphic to a given irreducible submodule of
called a homogeneous component of
Vi
is a nonzero homomorphism from
written as a sum of irreducible isomorphic modules. submodules of
with
i
are irreducible,
completely reducible R-module
A
58 V j
W zz Vi'
for which
W
is irreducible, and let W be an
V
is
V.
Let V = 0 Vi, where each Vi is irreducible and let J 5 I GI be such that V . j E J are the representatives of all isomorphism classes of Vi, j' i E I. For each j E J , denote by Xj the sum of all Vi with Vz.. E Vj' 3 . 6 . PROPOSITION.
Then The X
(i)
jE J
Proof.
V
are all homogeneous components of
if
(i) Apply Lemma 3.5.
lii) Direct consequence of the definition of X j. Let V
be any R-module.
Then a submodule W of
V is said to be fully
i n v a r k n t in V if it admits all R-endomorphisms of V. 3.7. LEMMA.
Let
v
# 0 be a completely reducible R-module.
is fully invariant if and only if of
submodule W of V
sum of certain homogeneous components
v. Proof.
Vi
W is a
A
We keep the notation of Proposition 3.6.
then either f c V i ) = 0
Vj,
Thus f ( X . 1
J
G Xi r
or f ( V . )
2
proving that each X
j'
If f E End(V) and R since Vi is irreducible
V. 2 V z j and hence the sum of certain X j' I?
is
DZRECT DECOMPOSITION OF RINGS
fully invariant.
w
Conversely, assma that
P + Q
= P,
= P @
P-+Q
:
Q and therefore V
J,
of
v
=
@
x
so
Let
V by
0 be a completely reducible R-module and express
of its homogeneous components:
sum
0,
=
P @ Q 0 U for some submodule 1! of V.
=
be the given isomorphism and define an endomorphism
vf
P is
V, we need on1yshow:if
If Q = P, this is clear; otherwise P n Q
then Q C_ W.
3 . 8 . COROLLARY. Let
To prove that
W and Q is an irreducible submodule of V such
an irreducible submodule of
f
W is fully invariant.
is a sum of certain homogeneous components of
that Q
15
V as the
Then
j € J j'
Proof.
(f.1
Any family
of endomorphisms of
X. defines an endormorphism 3
3
f of
V and it is clear that the correspondence
(f.)wf is an injective 3
homomorphism
Because each X
is fully invariant &emma 3 . 7 1 , any endomorphism f
i
into itself. each X j required.
Setting fj = flXj,j E J ,
it follows that
V
of (fj)
maps
*f
as
4. DIRECT DECOMPOSITION OF RINGS. Let
(Ri),iE I be a family of rings and let R be their direct product.
the natural projections Ti : R
xi
:
Ri
-+
R
+
Ri are ring homomorphisms, while the injections
preserve addition and multiplication, but not the identity element,
and so are not ring homomorphisms.
Of course the latter implies that the images
of the R
i are not subrings, although they are ideals in R. Let R ,R ,...lR
4.1. PROPOSITION. (i)
Then
If R = R
1
1
x
... x
2
n
be arbitrary rings
R n l then there exists a decomposition of 1 1
CHAPTER I
16
1= e
... + en
i-
a s a sum of pairwise orthogonal c e n t r a l idempotents such t h a t
R =Re 8 1
...,en
e,, n
If
(ii)
1 ei
1=
with
i=1
,
then
R
Re 0
=
1e
with
i=1 i
ei
. _ .8 Ren
I
z
R
(1 G
i
E
.
e ,. .,e
li, then
R
and
i G n) R
... x Ren .
Re x Re x 1
R such t h a t R
are i d e a l s i n a r i n g
n 1 =
Re.
and
are pairwise orthogonal c e n t r a l idempotents of a r i n g
Il,...,I
(iii) I f
... 8 Ren
= I
8
... 63 In
and
a r e pairwise orthogonal c e n t r a l idem-
n
R such t h a t I. = Re 1 G i G n. I n p a r t i c u l a r , R I x ... x In. i' 1 Proof. (i) L e t xi : Ri* R be t h e n a t u r a l i n j e c t i o n , and l e t e be t h e i image of t h e i d e n t i t y element of R 1 G i G n. Then e ,...,e a r e pairwise i' n n orthogonal c e n t r a l idempotents of R with 1 = 1 E Hence R = Re 63 . .. 0 Ren i=1 i' Since I d . = Rei and ei i s t h e i d e n t i t y element of t h e r i n g Rei, t h e r e q u i r e d
p o t e n t s of
a s s e r t i o n follows. The f i r s t statement i s obvious.
(ii)
P
*
(re
,...,rn
r
any
,...,re,) in
The map
R
4
Re x 1
... x Ren,
is obviously an i n j e c t i v e homomorphism of r i n g s .
R, r e + 1 1
.. . i- rnen
Since f o r
(riel,...,rnen ) ,
i s mapped t o
t h e map
i s an isomorphism.
n
i
(iii) I f
#
j, then
e.e. E I . n I . = 0 and so e R
Hence, by Proposition 2 . 1 2 ,
Ii = Re. z
=
e.R,1 G i z
z
1-3
n.
=
3
Re 8
...
j 8 Ren = e R 8
I t follows t h a t
1
zei
=
e x
1-e j
=
=
=
1 eiej
... 8 enR z
for all
=
.
e?
3 and t h e r e f o r e
i=l
3c E
Ii,
I.I. c I . n I . = 0 f o r i # j , w e deduce t h a t ei i s a z 3 - z 3 The l a s t statement i s a consequence of (ii) c e n t r a l idempotent, 1 4 i G n. 1G
i (n.
Since
.
A ring
R,R
=
R i s c a l l e d indecomposabZe i f f o r any i d e a l s X and Y of
X @ Y implies X
= 0
or
Y = 0.
The following c o n d i t i o n s a r e equivalent.
4.2.
COROLLARY.
(i)
R is indecomposable
(ii) (8) i s indecomposable (iii)R i s not isomorphic t o a d i r e c t product of
(iv) 0
and
1 a r e t h e only c e n t r a l idempotents of
2
R
rings
m
17
MATRIX RINGS
Direct consequence of Proposition 4.1.
Proof.
Assume that R
4.3. PROPOSITION.
*,...,I n
is either artinian or noetherian. such that R = I
of R
exist ideals 11,1
8
... 0 In
Then there
and each Ii
is
an indecomposable ring. Assume that R
Proof. R = I
1
is noetherian.
We may choose a direct decomposition
@ I ‘ in which 1; is (proper) maximal, in which case Il will be 1
Next write I ’
indecomposable.
1
=
I
2
@ 1’, where 2
I
# 0
and I‘
is maximal.
Continuing in this way, we obtain a sequence of decompositions R = 7
o ... 0 I n o r;
Because the associated sequence of ideals II @
... @ I n
terminate, we deduce that I ’ = 0 at some stage.
n
choose In minimal at each stage, then I;
( n = 1.2,
In case R
...)
must
is artinian, we
is a descending chain which must
So in either case we obtain the desired decomposition.
again stop.
4.4. COROLLARY.
Any artinian or noetherian ring is isomorphic to a direct
product of finitely many indecomposable rings. Apply Propositions 4.1 and 4.3.
Proof. 4.5.
Let R ,R
COROLLARY.
Then R
1
,...,R
2
be arbitrary rings and let R = R
is artinian (noetherian) if and only if each Ri
Proof.
X
... x
R.,
is artinian (noetheriad
Direct consequence of Proposition 4.1 and Corollary 2.8.
5. MATRIX RINGS.
be an arbitrary ring and let S = Mn(R).
Let R
..
consisting of all scalar matrices diag (r,. ,r),P E R .
its image in S
every ordered pair entry
1 and
We shall identify R
( i , j ) ,1 < i , j
< n,
let e . .
The elements e i j ,
0 elsewhere.
called the matrix u n i t s ,
satisfy the following properties (i)
eijeks = 0
(ii) 1
=
e
11
+
if j
f
k
and eijeks
=
eij
... + e nn
(iii) The centralizer of
Ie..}
L I
in S
is R.
For
be the matrix with ( i , j ) - t h
23
if j = k
with
18
CHAPTER I
Observe also that R
1
e
S e
11
.
11
(i), (ii) and (iii) determine S 5.1. PROPOSITION. {V..I
23
i,j
1
set in
s.
s
Let
n)
The following simple observation shows that
up to isomorphism.
be a ring that contains a set of elements
satisfying (i) and (ii) and let R
be the centralizer of this
Then the map
Furthermore, R
is an isomorphism of rings and R-modules.
E
v
11
ull*
We know that M (R) is a free R-module with the matrix units
Proof.
Hence e i j
as a basis.
clearly coincides with
++
vij
$.
e.. '3
extends to a homomorphism of R-modules, which
In view of (i), (ii) and Iiii),
Q is easily seen to
be a ring homomorphism. Assume that
t
E
11,
...,nl
1.aijUij
= 0 and let k , s E {1,2,.
i r3
( - 1f i j V i j )
0 = Vtk
vst
r3
=
.. ,n) .
Then for all
ak s v tt
and so
Finally, fix s E
s
and for each i , j P
Then for all Opt
set - v s v ki jk
ij -
we have UPtYij
= pptVkiS
Vjk =
upis v j t
and P { jU p t =
Thus rij foregoing
commutes with a l l
upt
v k i s 'jk
= 'pis
'jt
and therefore P i j E R.
Furthermore, by the
WTRIX RINGS
proving that $
e
11
Since $ ( e l l Mn(R)e
is surjective.
M (R)ell R ,
19
u SO
) = 11
and since
11
11
the result follows.
n
and a positive integer n, we write
Given a (left) R-module V V.
n-th direct power of
v"
for
The following result shows that matrix rings arise as
endomorphism rings of direct powers of modules.
5.2. PROPOSITION.
End($) R
1 Mn(End(V))
R
Given i , j E {1,2,.
Proof.
,.. . ,On) =
EijW
Then $(Ul
E..E
ks
23
,...,V n )
if $
= 1
$
=
1
1
Eij E End(?)
=
by
R
..
(0,. ,vj,o
t i +
,O)
I . . .
and write
1.
,...,
... = qn.
=
define
Let $ E End($) R Qn(On)). Then $ E =~ E..$ ~ -L3 Hence the centralizer of { E .
and
= ($ (U )
2
End(V).
6. J kEis
.. ,n),
for all i,j if and only
.I
is identifiable with
23
Now apply Proposition 5.1.
R
$ as an R-module.
In the preceding discussion we treated consider
v"
elements of
Ax
as an M (R)-module in the following natural way.
n
fl
as column vectors and, for each A E Mn(R)
as the matrix multiplication.
5.3. PROPOSITION.
(i) If
w
and x E
$, define
V,
is a submodule of
then the map
v
b/w W n
onto the lattice of
vM
submodules of End(?)
End(V) R
M, (R) (iii) The map
VC+
ism classes of R given by
We visualize the
With this convention, we now prove
provides an isomorphism from the lattice of submodules of
(ii)
But we can also
?
induces a bijective correspondence between the isomorph-
and M (R)-mcdules.
The inverse of this correspondence is
I J k ell W.
Proof.
(i) The correspondence
Wk w"
is clearly order-preserving.
Therefore we need only show that it has an inverse which is also order-preserving. Consider the projection on the first factor
7T
:
fl+ v
and, for any submodule
20
CHAPTER I
v",
X of
preserving.
f
fl = x,
) =
is order
From the action of Mn(R)
?
=
on
for a submodule 'L
of
proving the assertion.
and fix f E End(?).
Put S = Mn(R)
Then, by the nature of action of
S
v",f
on
(gf)(W) = W.
can be written in the form
(fg)(X) = f ( V
Hence
(ii) S
It is evident that
we see that X
V.
Then the correspondence X * g ( X )
put g ( X ) = TI1 (XI.
has the same projections, say Xf,
any $ E End(V)
on all factors.
determines an element of End(?)
R
Conversely,
whose projections on all
S
factors are equal to $.
Hence the map
f* A
provides the desired
f
isomorphism. (iii) It suffices to prove that e are R
e
?= 11
11
W
1
?
and Mn(R)-modules, respectively. V x 0 x
+
... x
0
V.
Since eklw = 0
modules.
e
11
... + eni wn
2-isomorphism of
f C-+
V
and
11
Note also that
1
,...,e 11Wn )T
W onto (ellW)n .
11
W
=
e
11
W
W = 0,1
... 0 enl
k
n,W E
.
V
and
' W as R-
W, the map
is well defined and is at least an
We are therefore left to verify that
23
latter being a consequence of the action of Mn(R) on follows.
11
@
f ( e . . e w ) = e . . f ( e k l o ) for all w E W and all i , j , k E {1,2 ZJ k i
where
By the definition of e
if and only if e
(e W
w,
(ell
,...,n].
The
( e l l W ) n , the result
Given an ideal I of R, we write
5.4.
PROPOSITION.
ideals of R
(i) The map
Proof.
Mn(I) is a bijection between the sets of
and Mn(R).
M, ( R )/Mn (I)
(ii)
I+
Mn (R/I)
(i) It is clear that M (I) is an ideal of Mn(R) and that the
given map is injective. entries of elements in J .
Let J
be an ideal of Mn(R) and
Then I is an ideal of R
I s R consist of all
such that J = Mn(I).
The natural map R - PR/I induces a surjective homomorphism
(ii)
Mn(R)-+
Mn(R/I)
whose kernel is Mn(I).
For future use, we next record
kr
MATRIX RINGS
5.5.
21
Let V be an R-module and let e be an idempotent of R.
LEMMA.
Hom(Re,V)
eV
Then
as additive groups
R Similarly, if Proof. f h - 4
V is right R-module, then Hom(eR,V) R
f e Hom(Re,V),
then ef(e) = f(e2) = f(e) E eV. Hence the map R is a homomorphism from Hom(Re,V) to eV. Conversely, if v E eV
f(e)
If
Ve
R then the mapping g
2,
map
2)-
:
xe*
is an R-homomorphism from Re
XG
is an inverse of f++ f(e),
g,
to
V.
Since the
the first isomorphism follows.
The
second isomorphism is proved by a similar argument. 5 . 6 . PROPOSITION.
Let e be an idempotent of a ring R. End(Re) R
as rings
and End(eR) a eRe R
(eRe)'
g
Then
In particular End(#) R
3
Ro
and End(R
) g
R
R R
Applying Lemma 5.5 for V = Re, we see that the map f-
Proof.
f(e)
is
anisomorphism of the additive group of End(Re) onto the additive group of eRe. R Given f,g E End(Re), write f(e) = er e and g(e) = er e for some r1,r2E R. R Then
(fg) (e) = f(er e) proving that fhf(e)
=
er er e = (er el (er e) = g(e)f(e), 2
1
reverses the multiplication.
Since e
is the identity
element of the ring eRe, the above map preserves identity elements. proves the first isomorphism.
This
The second isomorphism is proved by applying
Lemma 5.5 for V = eR. 5.7.
LEMMA.
(Schur's lemma).
Let
V be an irreducible module.
Then End(V)
R
is a division ring. Proof. Let f : V + nonzero submodule of and thus Kerf
=
0.
V be
V, f(V)
=
a nonzero R-homomorphism.
V.
Since f(V)
is a
On the other hand, since f # 0 , Kerf #
Hence f is an isomorphism, proving the assertion.
V
22
We are now ready to prove 5.8. PROPOSITION.
Assume that
&?
is completely reducible.
artinian and there exist primitive idempotents e ,e $ 1
integers nl , T I 2 , .
..,nr
,...,er
of R
is
and positive
such that v
R
(eiRei)
iJMn
2
i
i=l
and each ez3ei
2
Then R
is a division ring.
Proof. Since 1 lies in the sum of finitely many irreducible submodules of R R , the same is true for
#.
. . ,en}
exists a set { e l , .
#
Rel @
=
... @ R e n ,
Hence R
is artinian, so by Corollary 2.13, there
of pairwise orthogonal primitive idempotents of R with
where each Rei
...,Rer
We may assume that Re 1 ,Re 2 ,
is indecomposable and hence irreducible.
are all nonisomorphic among the Rei,l G i G n.
. is a homogeneous component of &? corresponding to Rei,
If X
(Proposition 3.7Cii1 1 and X
i
Applying Propositions 5.6,
and therefore R
2
(Rei]
ni
then
fl = 2:
@
...@ Xr
i f1 G i G r.
for some positive integer n
5.2 and Corollary 3.9, we derive
nMni (eiRei).
The fact that eiRe
i=1
.
i is a division rirg being
a consequence of Lemma 5 . 7 and Proposition 5.6, the result follows. 6. THE RADICAL AND SOCLE OF MODULES AND RINGS.
V be a module over an arbitrary ring R .
Let
Then the radical of
J ( V ) , is defined to be the intersection of all maximal submodules of case
V contains no maximal submodule we set J ( V )
J(R)
of R
We say that R
If V
socle of R ,
In
The Jacobson radical
is the intersection of all maximal left ideals of R .
is semidmple if J(R1 = 0.
The sum of irreducible submodules of
V.
V.
is defined by
Expressed otherwise, S(R1
of
= V.
V , written
8, written SocV, is called the socle
contains no irreducible submodules, then we set SocV = 0.
written S o d ,
is defined by
S o d = Soc(#1.
The
THE RADICAL AND SOCLE
6.1. PROPOSITION. (i) If
(ii) If
f
(v)
0
f
f
V.
f
V.
is a finitely generated R-module, then J ( V )
is an artinian module, then Soc(V)
f
0
V is completely reducible, then J ( V ) = 0
(iii) If (iv) V
0
V
23
is completely reducible if and only if
SocV
=
V
SocV is the unique largest completely reducible submodule of Proof.
V
(i) By Proposition 2.5,
(ii) By Proposition 2.1,
V
V.
has a maximal submodule and so J ( V )
has an irreducible submodule and thus Soc(V)
(iii) We may clearly assume that V
in which case V
0,
f
f
0.
is the direct sum of
Hence, by the definition of J ( V ) ,
irreducible submodules (Proposition 3.2). J ( V ) = 0.
(iv) Again we may assume that (v)
f
0.
.
Now apply Proposition 3.2.
Direct consequence of Proposition 3.2.
6.2. PROPOSITION.
If
V - f .0
is an R-module, then the following conditions are
equivalent: V
(i)
is a direct sum of finitely many irreducible modules
(ii) V
is artinian and completely reducible
(iii) V
is artinian and J ( V )
=
0
Proof. The implication (i) * (ii) is a consequence of Proposition 3.2 and Corollary 2.8, while the implication (ii) * (iii) follows from Proposition 6.1(iii). Assume that (iii) holds. Since J ( V ) = 0 , V
,...,V n
By Proposition 2.1,
this implies that
n n V.
=
0
V.
of
Hence
f (J(W)
V
(V/Vi)
and so
V
Now apply Corollary 3.3.
be a homomorphism of R-modules.
with equality if f is surjective and
Then
Kerf C_ J ( W )
socv
(i), If
5 J(V).
V has no maximal submodules, then J ( V )
Assume that M
is a maximal submodule of
f* given by
nn
is isomorphic to a submodule of
Let f : W +
5J ( V )
(ii) f(S0CW)
f(J(W))
for some maximal submodules i=1
6.3. PROPOSITION.
Proof.
is finitely cogenerated.
i=1
is completely reducible, by Proposition 3.1(ii).
(i)
v
:
W+
V/M
= V
V.
and hence Then the map
CHAPTER I
24
is an R-homomorphism.
Since Kerf*
and therefore f ( J ( W ) C_ M .
Thus f ( J ( W ) 1
J(V).
is surjective and Kerf C_ J(W).
Assume that f
submodules, then so does V ,
If
W has no maximal
in which case
Assume that the set { M . l i E I } of all maximal submodules of z By hypothesis, Kerf
C Mi
for all i E
W containing Kerf
submodules of
U M f ( U ) ,U E X,
W,J(W) CKerf*
is a maximal submodule of
I.
and submodules of
is an isomorphism of X
set of all maximal submodules of
If X and
V
onto
Y.
V,
Y
W is nonempty.
are the lattices of respectively, the map
Thus
{f(Mi)
E
I} is the
and
as required. Iii) Direct consequence of Propositions 3.l(ii) and 6.l(v). 6.4. COROLLARY.
(i)
Let V
be an R-module and let
W be a submodule of
v.
C J(V)
J(W1
2 (J(V)+W)/W W 5 J C V ) , then
(ii) J(V/'/w) (iii) If
(iv) SOCW = Proof. where f
J(V/W) = J ( V ) / W
W n SOCV (i) Direct consequence of Proposition 6.3(i) applied for the case
is the inclusion map
(ii) and (iii) Apply Proposition 6.3(i) for the case where f : V
4 V/W
is the
natural homomorphism (iv) By Proposition 6.3(ii), applied for the case where f we have
SocWC SocV and hence SocW 5
W n SocV.
i s the inclusion map,
a consequence of Proposition 3.2(ii), and 6.l(v), the result follows. 6 . 5 . COROUARY.
Let V be an R-module.
for any submodule IJ of
V
Then J ( V / J ( V ) ) = 0
for which J ( V / W ) = 0.
.
The opposite inclusion being
and J ( V )
5W
25
THE RADICAL AND SOCLE
Proof.
.
The first equality is a consequence of Corollary 6.4(iii), while the
second follows from Corollary 6.4(ii). 6.6. COROLLARY.
W
(i)
=
J(V)
Let
V be an R-module and let W be a submodule of V .
if and only if
(ii) W = SocV if and only if Proof.
W C_J(V) and J ( V / W ) W
2. SocV
and
SocW =
=
0
W
(i) Direct consequence of Corollary 6.5
(ii) Direct consequence of Proposition 6.1 (v).
6.7. PROPOSITION.
Proof.
Let
(Vi),i E I be a family of R-modules.
(i) We may clearly assume that each
Then the product M i x
nVj j+i
Vi
Then
has a maximal submodule M
i'
is a maximal submodule in
V = nVi.
Hence J ( V )
iEI
is contained in each product JCVi) x n V j
and the intersection of these is
j+i nJ(Vi),
whence the assertion.
iEI
(ii) By the argument of (i), Mi 8
so that J ( S )
i
E
I.
C
i
:
W-+
8 V.)
@ J(Vi). iEI
s=
is a maximal submodule in
j+i 3 But, by Proposition 6.4(i),
J(vi)
5 J(s)
@ Vi iEI
for each
Hence the opposite containment is also true and the assertion follows.
(iii) We may clearly assume that 71
(
V = 8 V
has an irreducible submodule W.
i V. is the projection map, then W 5 fB ni(W).
Now each
either 0 or irreducible.
Thus
Ti(W)
5 SocVi
is
Ti(W)
ieI and therefore SocV
5
If
@
SocVi.
iEI
The opposite containment being a consequence of Proposition 6.l(v), the result follows.
' For any R-module V, J ( V ) = 0
6.8. COROUARY.
if and only if
V is isomorphic
to a submodule of the direct product of a family of irreducible R-modules.
tVi) ,i E I
Proof. Let
Proposition 6.7(i),
be a family of irreducible R-modules.
J ( n V i ) = 0.
By
Hence the "if" part is a consequence of
iEI
Corollary 6.4.
Conversely, assume that J ( V ) = 0.
nothing to prove.
Hence we may assume that
V
# 0,
If V = 0
then there is
in which case there is a
26
CHAPTER I
of maximal submodules of V
(Wk),k E K ,
nonempty family
V admits an injective homomorphism into
n (V/Wk).
n Wk = 0. MK
with
Thus
V/Wk is
Since each
KK irreducible, the result follows. submodule W
A
x
of
4
V is called essential in V in case for every submodule
v,
of
W n X = 0 implies
Dually, a submodule W
of
X
=
0
V is called superfluous in V in case for every
X of V
module
W +X 6.9. PROPOSITION.
Let V
=
V implies X
=
V
be an R-module.
v
is
is the unique largest superfluous submodule of
V.
(i) J(V) is the sum of all superfluous submodules in finitely generated, then J(V)
v.
Furthermore, if
(ii) SocV is the intersection of all essential submodules in if
5 J(V),
K + W of
V.
Furthermore,
V is artinian, then SocV is the unique smallest essential submodule of V. Proof. (i) Let W
W
sub-
=
be a superfluous submodule in
we may assume that V
V and hence K
=
module of
V with
module K
of
occw.
RX iW =
V , then either
implies that RX
.
If W E K, then
Thus every superfluous submodule
On the other hand, let x E
V with W S K and x 9 K
Thus x E J ( V )
To prove that
has a maximal submodule K.
V, a contradiction.
V is contained in J ( V 1 .
V.
w=v If x
v.
If
w
is a sub-
or there is a maximal sub-
s J ( V ) , then the latter cannot
is a superfluous submodule of
v,
proving the first statement. Now assume that V verify that J ( V ) submodule of But
is finitely generated.
is a superfluous submodule of
V with
v = w + J(V).
By the foregoing, we need only
V.
So assume that W
Then, by Corollary 6.4(ii),
V/W is finitely generated, so by Proposition 6.1(1) v/W
=
0.
is a
J(v/w) = v/w. Thus
w=v
as required. (ii) Let M
be the intersection of all essential submodules in V .
To prove
THE RADICAL AND SOCLE
that Socv EMI
V has an irreducible submodule W.
we may assume that
V, then L n W # 0 and so W
is an essential submodule of
and therefore we need only verify that M of M.
submodule N
N n N ’ = 0, then
N
5 M 5 N @ N ’,
This N
+
N’ = NO N ’
is a direct summand of
v
Thus SocV
Since V
M and therefore SocV
So fix a
V.
But then
=
M.
We need only verify that Socv is an
W n SocV
Assume that
is artinian, so is W
has an irreducible submodule X.
0 for some nonzero submodule
=
But then X @ SocV is a completely reducible
is essential in V, as required. 6.10. PROPOSITION.
Let
Hence SocV
’
V be an R-module generated by if and only if for all r
Then U E J(V)
W
and hence, by Proposition 2.1,
module strictly containing Socv, contrary to Proposition 6 . l ( v ) .
V.
5M
V maximal with respect to
is an essential submodule of
is artinian.
v.
essential submodule of
U
If L
and by modularity
Now assume that
W of V.
2L.
is completely reducible.
If N’ is a submodule of
N
27
i
E
U
,...,U n ,
R, 1 G i
f
and let
n, the elements
ui + r .u, 1 G i G n, generate V. Proof. Let U E J(V) elements U .
+ riu,
and let
i Q n.
1
W be the submodule of V generated by the W + J(V)
It is clear that
=
V so that W
= V,
by Proposition 6 . 9 ( i ) . Conversely, assume that U
V such that i
U
4 W.
there exists r
i
J(V).
Then there is a maximal submodule W
Hence we must have RU E
thus the elements U . z
R
such that ui E
+ r z.u,
1
Q
i
-
+W
=
+ W,
r.U
V.
of
It follows that for each
whence U i + riU
n , do not generate V.
W and
’
We now turn our attention to the radical and socle of rings. Let
V be an R-module.
It is clear that ann(V)
The annihiZator of
v, written ann(V),
i s an ideal and that
is defined by
V may be viewed as an R/ann(V)-
CBaPTFR I
28
module.
V is faithfd if ann(V)
We say that
An ideal
=
0.
I of R is called primitive if the ring R/I has a faithful Clearly I is primitive if and only if I
irreducible module.
is the annihil-
ator of an irreducible R-module. Let V
6.11. LEMMA.
V
1
R/X
t,
so V
=
Then V
is irreducible if and only if
for some maximal left ideal X of R.
Proof. let
be an R-module.
Clearly
R/X
be a nonzero element of RV.
The map
R
Suppose that V
is irreducible.
--+
V ,P
V. I-+
Then RV 19
is irreducible and
V
is a nonzero submodule of
is a surjective R-homomorphism.
R/XZ V for some left ideal X o$ R.
is irreducible, X
Since V
and
Hence
is
maximal. Every maximal left ideal contains a primitive ideal and every
6.12. LEMMA.
primitive ideal is the intersection of the maximal left ideals containing it. Let X be a maximal left ideal of R.
Proof.
Hence the annihilator of R/X
irreducible R-module.
By Lemma 6.11, R/X
is an
is a primitive ideal
contained in X. Let I be a primitive ideal in R whose annihilator is I . Clearly Mv and so Mv Mv
with
0 # v E V,
J(R)
5I
in V ,
Since V
is a maximal left ideal in R. the result follows.
put Mv = {r E Rlrv = 01.
is irreducible, V = Ru
R/M
.
Let R be a ring. In
is an ideal of R.
is the annihilator of a completely reducible R-module. if I
v
Since I is the intersection of all
is the intersection of the annihilators of irreducible R-modules.
particular, J(R) (ii) J(R1
Given a nonzero v
is a left ideal in R.
6.13. PROPOSITION. (i) J(R)
and let V be an irreducible R-module
Furthermore,
is the annihilator of a completely reducible R-module.
(iii) SocR is an ideal of
R.
Proof. (i) Direct consequence of Lemmas 6.11 and 6.12. (ii) If
v=
@ Vi, iEI
then ann(V) =
n ann(Vi).
Setting
CVi) ,
for some k
Setting I
1.
Assume by way of contradiction that I f 0 . with IX # 0, for example X = R. all such X .
Then I ( J M )
=
Let M
( I J ) M = IM # 0
we must have R x =
6.23. PROPOSITION. if I V # V
Let I
Jk
we have I'
and, since JM
5M
=
I.
of R
we have JM
=
M.
is a finitely generated R-
such that T x # 0 and so I ( R x ) # 0.
M, as required.
be a left ideal of R.
Then I E J I R )
if and only
for any nonzero finitely generated module V .
Proof. Let V # 0 be a finitely generated R-module. Hence, by Proposition 6.14, if I
Proposition 6.l(i). IV C - J(R)V
By hypo-
be a minimal element in the set of
By hypothesis, there exists x E M
As Rx E M
=
2 J 2 3 .. . .
There exist left ideals X
By Proposition 6.15, we are left to verify that M module.
is nilpotent.
and consider the chain J
Put J = J ( R )
Proof.
k thesis, J
is artinian, then J(R)
5J ( V )
rated module
V,
# V.
Conversely, if IV # V
6.24. PROPOSITION.
Let
(Ri) ,i E I ,
icI
An
element
n
is1
(pi) E
for all i E I .
6.25. PROPOSITION.
Proof.
For any ring R
Let V
then
V.
be a family of rings.
Thus I C J ( R )
Then
J(Ri)
n R i
.
is a left unit if and only if ri
is1 left unit of Ri
by
for any nonzero finitely gene-
.
J(nRi) =
Proof.
5J(R),
then IV = 0 for any irreducible R-module
by Proposition 6.13 (i).
Then J ( V ) # V
Now apply Proposition 6.18.
and any positive integer n ,
be an R-module and let
v"
be an
Mn (R)-module as in
is a
32
CHAPTER I Then ann(V7 = Mn(ann(V1),
Proposition 5.3.
by the definition of
Vn.
the desired conclusion follows by virtue of Propositions 5.3 and 6.13(i). PROPOSITTON.
6.26.
Let
R be a ring.
Hence H
Then the following conditions are
equivalent: (il
is a direct sum of finitely many irreducible modules
$
(iil
#
(iii)
R
i s completely reducible is semisimple artinian
(ivl Every R-module i s completely reducible (v) R
is a finite direct product of full matrix rings over division rings. The equivalence of (i) - (ivl is a consequence of Propositions 5.8,
Proof.
The implication (iil * (vl follows from Proposition 5.8.
and 3.4.
2.1,6.2
implication (v) * (iii) is a consequence of Proposition 6.25,6.24
and 5.3 together
with Corollary 2.8. A
ring R
is said to be s h p k if 0 and R
PROPOSITION.
6.27.
Let R
be a ring.
are the only ideals of R.
Then the following conditions are
equivalent. (i) R (ii
is simple artinian
&?
(ii
is a direct sum of finitely many isomorphic irreducible modules
R is isomorphic to a full matrix ring over a division ring
)
R is semisimple artinian and a l l irreducible R-modules are isomorphic.
(iv
Proof. Cil *
Wl: Direct consequence of Propositions 6.26 and 5.8
(iil * Liiil: Follows from Propositions 5.2 and 5.6 and the Schur's lemma. (iiil * livl: Follows from Propositions 6.26 and 5.3. (iv)
*
(1):
Follows from Propositions 6.26,5.2
As a partial converse of Proposition 6.14, 6.28.
PROPOSITTON.
Proof.
Let . V
By Proposition 6.14,
J(Rl V
and 5.4.
we prove
be an R-module.
If R/J(R)
5 JLV)
The
is artinian, then
and hence, by Corollary
THE RADICAL AND SOCLE
6.6(i), we need only show that J(V/J(R)V) (Corollary 6.17 (ii)) 6.26). Hence
, every
0.
Since J(R/J(R)1
=
0
R/J(R) -module is completely reducible (Proposition
In particular, the R/JCR)-module V/J(RIV
=
33
V/JCR)V
is completely reducible. = 0,
is a completely reducible R-module, and therefore J(V/J(R1V1
'
by Proposition 6.l(iii)6.29. PROPOSITION.
Let R/J(R)
be an artinian ring.
Then
SocR = { r E RIJ(R)r = 01 Proof. all x
E Sod?.
Since
V
Since SocR
is a completely reducible R-module, J(R1x
Setting V = {r E RIJCRIP = 01,
module (Proposition 6.2.61. by Proposition 6.1(vlI 6.30. PROPOSITION.
V
Proof.
V
Hence
is a completely reducible R/J(R)-
is a completely reducible R-module and so,
Let R be a ring such that every irreducible R-module is Then
The hypothesis on R
for every irreducible R-module
6.31. PROPOSITION.
5V.
5 SocR. '
isomorphic to a submodule of R R
PV = 0
V
can be regarded an R/JCR)-module,
it follows that SocR
0 for
=
Let R
ensures that rSocR = 0 if and only if
v.
.
Now apply Proposition 6.13 (i)
be an artinian ring and let V
8
be an R-module.
Then the following conditions are equivalent: (i)
[I
is of finite length
(ii) V (iii)
is artinian
V
is noetherian
Proof.
By definition, (i) * (ii) and (i) * (iii).
either artinian or noetherian.
By Proposition 6.22, J(RIn = 0
By Proposition 2.1, it suffices to show that the modules
.-.,J(R)n-lV/J(R)nV annihilated by
Assume that
have composition series.
for some
v YZ
is 1.
V/J(R)V, J(R)V/J(R)'V,
However these modules are
J(R) and so can be regarded as R/J(R)-modules.
Furthermore, by
Proposition 6.26, each of these modules is completely reducible and, by hypothesis,
CHAPTER I
34
.
each of them is either artinian or noetherian. fore a consequence of Corollary 3.3. Let R
6.32. COROLLARY.
(i) R
The desired conclusion is there-
be an artinian ring.
is noetherian
(ii) R R has a composition series (iii) Every finitely generated R-module is of finite length. Proof. (i) Apply Proposition 6.31 for V = R R (ii) Apply (i) and Proposition 2.10 (iii) Let V be a finitely generated R-module.
Since
image of a direct sum of finitely many copies of R, V
V
is a homomorphic
is artinian.
Now apply
Proposition 6.31. Let R
6.33. PROPOSITION.
be an arbitrary ring, let V
be an R-module, and let
S be a subring of R. (i) If
V
has a composition series V 2 V 1
2
3
...
In particular, J(Rln = 0 if
then J(RInV = 0.
3
V
Vn+l 3 0
of length n ,
is a faithful R-module.
(ii) If every irreducible R-module is completely reducible as an S-module, then
J(S) 5 J(R). annihilates each factor Vi/Vi-l,
Proof. (i) Because J(R)
J ( R ) n must
annihilate V . (ii) Let
xE
J(S)
and let V
be an irreducible R-module.
.
completely reducible as an S-module, xV = 0. assertion follows.
Hence z E J(R)
Since V
is
and the required
The following consequence is a sharpened version of Proposition 6.22. 6.34. COROLGARY.
length of
RR
Proof.
Let
d be an artinian ring and let n be the composition
Then
By Corollary 6.32,
Proposition 6.33 for
V
=
RR
R R has a composition series.
Now apply
35
THE RADICAL AND SOCLE
Let e
PROPOSITION.
6.35.
0 be an idempotent of a ring R .
f
Then
J(eRe) = eJ(R)e Proof.
Let I be a primitive ideal in R Then eV
irreducible R/I-module.
J(eRe) 5 eRe
5 I.
and so
eV = W .
Let 0 c W = eW
It follows that eV
5 I.
Since I was an arbitrary primitive
Then eue E J ( R )
.
and so, by Proposition 6.18,(1-b)
in which case beae = eue+b.
on both sides yields eue+ebe = ebeue.
Kence
e
Multiplying by
(e-ebel (e-eue) = e
Since eJ(R)e is an ideal of eRe,
is a left unit.
W is an
and so J(eRe) = eJ(eRee1e C_ eJ(R)e
this yield8 J(eRe) C - JCR)
(1-eue) = 1 for some b E R ,
where
is an irreducible eRe-module whence
Thus J(eRe1
Suppose that u E J ( R ) .
5 eV
Hence eV = eRe Kf C_ W
RW since V is irreducible.
J(eRe)V = J(eRe)eV = 0 . ideal in R,
If eV = 0 then
is an eRe-module.
Assume that eV # 0.
Then V =
eRe-module.
and let V be a faithful
so that e-eae
Proposition 6.18 implies
that eJ(R)e 5 J(eRe1. For any idempotent e
6.36. ~RoPosITIoN.
Proof.
Since
R
of R ,
J(Re) = J ( R ) e .
R = Re @ R ( l - e ) , we have J ( R ) = J(Re) @ J ( R ( 1 - e l )
(Proposition 6.7 (ii)) and J ( R 1 = J @ ) e @ J W ) (1-el
JW)e
5 J(Re)
and J ( R ) (1-e)
E R
But, by Proposition 6.14,
Hence J(Re) = J ( R ) e , as required.
Let I be a nilpotent left ideal of a ring R,
6.37. PROPOSITION.
x
5 J(R(1-el).
-
be a nonnilpotent element such that x2-x E I.
contains a nonzero idempotent Proof.
y such that y-x
k = 0, and set
Let
If z1 # 0 ;
and we are done.
z
1
let x
= x2-x E I. 1
commute with each other, and thus if x Hence x
a contradiction.
1
"hen the left ideal Rx
I.
d
= x+z -2xZ
and let
If Z l = 0, 1
E Rx.
choose y =
Then X,X
is nilpotent, then so is X
is a nonnilpotent element of RX,
and 3
X
1
-2
2,
Z
+2XZ
1
and direct
calculation shows that
r 2
1
The element z
2
=
x 12
-x
1
-
2
i
1
423 1
- 322 1
is nilpotent, contains z 2 as factor, and canmutes
1)
CHAPTER I
36
with x
1
.
Continuing in this manner, we may construct a sequence { xz. ) of non-
i
nilpotent elements of R x , choose i
x.
so that 2i 3 k ,
=
.
Moreover, xi # 0
= 0.
x. is the desired idempotent.
If we since
Let I be a nonnilpotent left ideal in a (left) artinian ring
6.38. COROLLARY.
8.
z - x z.
then we have X?
Thus y
is nonnilpotent.
occurs as a factor in x i2- x i .
such that 2'
I contains a nonzero idempotent.
Then Proof.
Let X
be a minimal member of the nonempty set of nonnilpotent left
ideals contained in I.
Then I 2
5I
and so I 2 = I.
Now let Y
in the set of left ideals contained in
x
such that XY # 0 ,
so that Xa # 0.
XU
5Y,
Then X*Xa # 0
and
k
and so a = x a
for some x E X
the nonnilpotent element x.
for all k
Invoking Proposition 6.37, we see that RX
so does I ,
since
0.
Y
=
and choose a E
Xa.
Y
Thus a = m
It follows that Y contains
Setting
Moreover, xa = Y # 0
we have xz-x E L .
whence
be minimal
I 3 X 2 Rx.
so
L cX
and thus L
is nilpotent.
contains a nonzero idempotent, whence
m
7. THE KRULL-SCHMIDT THEOREM
This section deals with the uniqueness of direct sum decompositions.
The main
result (Theorem 7.5) is Azumaya's generalization of a classical theorem of K?zull and Schmidt (Corollary 7.6). A ring R
is said to be a locaz ring if R / J ( R )
what follows, we write 7.1. LEMMA.
(i) R (ii) R
is a division ring.
In
U ( R ) for the group of units of R.
The following conditions are equivalent:
is local has a unique maximal left ideal
(iii) J ( R ) = R-U(R) (iv) The set of nonunits of R
Proof.
(i) * (ii):
Let
is a left ideal
I be a maximal left ideal of R.
Then I / J ( R )
is
37
THE KRULL-SCHMIDT THEOREM
and hence I = J(R).
a proper ideal of R/J(R)
be a unique maximal left ideal of R.
(ii) =) (iii): Let J
Since J(R)
intersection of all maximal left ideals of R, we have J(R) = J , an ideal.
The inclusion J(R)
If RX # R, then RX
so x E J(R).
Z E
(iii) (iv)
=)
=)
of R, fore
and so J
being obvious, assume that x
lies in a maximal left ideal of R, whence Rx then YX
On the other hand, if RX = R
Clearly y Q J(R), some
5 R-U(R)
1 = yx E J W .
otherwise
R and hence z
=
I.
=
E
is
R - U(R).
5J W ) ,
1 for some y E R.
Hence Ry = R,
Thus x E U(R),
is the
so zy = 1 for
a contradiction.
(iv) Obvious (i): Let x Q J(R) then M
x + J(R1
and let I = R - U(R).
I # R, whence M
=
I = J(R).
is a unit of R/J(R).
If M
is a maximal left ideal
Hence x
is a unit and there-
Thus R/J(R)
is a division ring, as
required. 7.2. COROLLARY.
Let R
Then the only idempotents of R
be a local ring.
are
0 and 1. Let e # 1 be an idempotent of R.
Proof. by Lemma 7.1, 7.3. LEMMA. 0
and
e E J(R1. Let V
Then e
is a nonunit and hence,
Now apply Proposition 6.18.
be an R-module.
1 are the only idempotents of
V is indecomposable if and only if
Then End(V).
In particular, by Corollary 7.2,
R if EndCV)
is local, then
V is indecomposable.
R Proof. projection
Assume that
V
=
X @ Y is a proper decomposition.
V - + X is an idempotent of End(V) R
Then the
distinct from 0
Conversely, assume that f # 0,l is an idempotent of End(V).
and
1.
Then V=f(V) $Kerf
R is obviously a proper decomposition. 7.4. LEMMA. (il
If
Let R
be a ring and let V
be an R-module of finite length.
V is indecomposable, then every f E End(V) R
is either a unit or nil-
potent. (iil
V
is indecompsable if and only if EndlV) is a local ring.
R Proof.
(i) We first show that an R-homomorphism f : V-+
and only if it is surjective.
Assume that f
is injective.
V
is injective i f
Then there exists
38
CHAPTER I
n
an i n t e g e r
such t h a t
u E V,
Therefore, f o r each
f
Since
f ( U ) = u E f(V)
is injective,
f
Now assume t h a t
U E V
there e x i s t s
f
and so
is surjective.
f n ( u ) = fn+'(U).
such t h a t
is s u r j e c t i v e .
By hypothesis, t h e r e e x i s t s an i n t e g e r
n
such t h a t
0 c Kerf c
f
Since form
fn)
(and hence
f o r some
U = fn(U)
U E Kerfn+l = Kerfn,
... c K e r y
= KerP+l
u E Kerf
i s s u r j e c t i v e , each U E V.
and thus
f(u) = 0,
Because
u = 0.
may be w r i t t e n i n t h e it follows t h a t
f
This proves t h a t
is injective.
I t w i l l next be shown t h a t
V
Since
i s indecompsable, t h e r e q u i r e d a s s e r t i o n w i l l follow.
n 2 1
integer submodule
fnCV)
injective. u E V
each
V
so
=
=
fn+l(V).
f
Then t h e r e s t r i c t i o n of
t o the
i s s u r j e c t i v e , so by t h e above t h i s r e s t r i c t i o n i s a l s o
This shows t h a t there e x i s t s
f"(V) @ Kerfn,
(ii) Assume t h a t
V
U E V
K e r f n = 0.
Because
fn(u - f n ( U ) )
such t h a t
fn(V)
= 0.
=
f2n(V),
E
i s a local ring.
E.
Then
Let
gf
1-gf i s a u n i t of
by v i r t u e of ( i ) . Thus
I t follows t h a t
w e deduce t h a t
E
E = End(V). R
f
By Lemma 7 . 3 , w e
be a nonunit of
E
and l e t
is a nonunit and hence i s n i l p o t e n t ,
E,
so by Proposition 6.18, f E J ( E )
is local.
W e a r e now ready t o prove
7.5.
THEOREM.
Let
R
for
a s required.
be an a r b i t r a r y element of
Invoking Lemma 7.1,
fn(V) n
is indecomposable and p u t
need only v e r i f y t h a t
g
fnCV)
such t h a t
Choose an
be a r i n g and l e t
V
be an R-module.
Suppose t h a t
39
TBE KRULL-SCHMIDT THEOREM
i,j
where f o r a l l
the rings
End(V.1
and
End(W.)
R Z V.
and a f t e r a s u i t a b l e rearrangement Proof.
rn
W
2
i
t h e p r o j e c t i o n s of
V
n @ W.. j=1 3
=
ei,fj
e f.e
are a l l i n
1 3 1
End(V1).
t h e r e s u l t i s c l e a r s i n c e by
= 1,
Vi,Wj,
onto
e
Because
R
= 1
e f.e
l o c a l , it follows from Lemma 7 . 1 t h a t
respectively.
n
c
j=l
V
morphism of
= e (V).
1
.-I
W e now prove t h a t
x
V.
(X)
= 0
e (y) = 0
as
e
Cf
e
)
(y) = 0 ,
and
f o r some
End(V1)
elfle,
j.
i s an auto-
f
= (f e ) (y)
X
V1 i s
on
y E 8.
f o r some
I 1
elf,el
f (V
proving t h a t
V
i s an automorphism on
.
W.
I t follows t h a t
1
)
n ( Q Vi)
Assume t h a t = 0. i=2 Hence e (U) = (e f e ( W ) f o r )
1
el(v) E e ( V ) = (e f e
Then W E
R
m
I
some
is
End(V1)
i=2
1
then
I 1 V E
and 1
f , (V1) 5 IJ1 and t h e k e r n e l of rn V = f (V @ ( 8 Vi).
Thus (elflel)(y) = el(x) and 2 =
3
Then
R
i=2
Hence
1
Thus
m fl(V1) n 8 vi,
E
e f .e
is a unit i n
1 3 1
Changing n o t a t i o n , i f necessary, it may be assumed t h a t
If
Using induction
i s indecomposable.
Denote by
0.
m = n
Then
...,r n } .
E {1,2,
rn V = @ V.
n
= 1
are local.
3
for a l l
i
or
observe t h a t i f
V
Lemma 7.3
and
z
We may harmlessly assume t h a t
on min{rn,n},
e
R
(V).
1
1
e (v-(f e 1
) 1
(W)l= 0
1
1
U - ( f e 1 (w) E
and
1
1
8 Vi. i=2
Thus m
m proving t h a t Because Lemma 7.3,
V =
f (V 1
Wl
f,(Vll 8 1
5W ,
)
8 Vi). i=2 t h i s implies t h a t (
i s indecomposable.
V
I
and
and hence, by i n d u c t i o n ,
i
E {I,
7.6.
V = W @ 1
rn = n
(
m =
{wl n 8 V i } . i= 2 w e see t h a t W1 =
f, (Vl) 8
f (V ) # 0 ,
Since
1
m i s isomorphic to
w
8 Vi). i=2
1
By
fl(V1)
The conclusion i s t h a t
and a f t e r a s u i t a b l e rearrangement
V.
Wi
...,ml.
COROLLARY.
(Krull-Schmidt theorem).
Let
V
be a nonzero R-module of f i n i t e
m length.
for
Then
V
can be w r i t t e n a s a d i r e c t sum
V = 8 Vi, i=1
where
Vi
are
CHAPTER I
40
n indecomposable submodules.
for each
V
= @ W
is another decomposition of j=1 j and (after possible reordering the W . ) we have Vi Wi 3
this kind, then n = rn
Moreover, if
i. Apply Proposition 2.9, Lemma 7.4Cii) and Theorem 7.5.
Proof.
.
8 . PROJECTIVE, INJECTIVE AND FLAT MODULES.
Let
R be
a ring.
A
sequence of R-modules and R-homomorphisms
v1 is emct at
Vi
-c
2
J1
J 2
v
v2
if Kerf. = Imfi-,
.
c
--t
... 'n-1
n'
The sequence is emct if it is exact at
each place, that is Kerfi = Imfi-, A short exact sequence is an exact sequence of the form
0-x-Y a
(*I Exactness of
(*)
KerB = Ima and
- Bz - 0
is equivalent to the following three conditions: a
8 is surjective.
The short exact sequence
In this case
B
induces an isomorphism
is split if a ( X ) is a direct summand of
(*)
is injective
Y.
This
is equivalent to either of the following: (i) There exists a y E HOm(Y,x) R identity map on X. (ii) There exists a
such that ya = lX, where
6 E Hom(Z,Y)
such that
86
=
lX denotes the
lZ
R The homorphisms y
and
6
are called splitting homomorphisms.
A diagram of R-modules and R-homomorphisms
."x
is conmutative if y = Ba.
Y
The same terminology applies to more complicated
PROJECTIVE,INJECTIVEAND FLAT MODULES
41
diagrams. Let V
be an R-module.
v
Then
is called projective, if for any given
diagram with exact bottom row
V
there exists an R-homomorphism y 8.1. PROPOSITION. (i) ~n R-module
The following conditions are equivalent
V
-
is projective
(ii) Any exact sequence 0 (iii) V
f3 = (Yy.
such that
B Y --+
X
V-0
of R-modules splits.
is a direct sueanand of a free R-module
Proof. ti) * (ii): Because
B
is surjective and
V
is projective, we have
the following commutative diagram:
V
Hence y
is a splitting homomorphism and (ii) follows.
(ii) * Ciiil: Take as
Y a suitable free R-module, and let
splitting homomorphism. (iii) * (i): Let V
Y
E : V-+
F
Then we can let
71
:
the natural injection so that
are R-modules and that
@ : V-
Y
surjective homomorphism, respectively.
-f
Y be a
is a direct summand of Y.
V
be a direct summand of the free R-module F ,
be a free basis for F . and
Then y(Vl
y : V
and
(Y
F+ TE
:
and let
{yi}
V be the natural projection = 1II
.
Now, suppose X and
X-+ Y are
a
homomorphism and a
We then have a natural map
to Y, and we can describe the situation pictorially as follows:
6T
from F
CHAPTER I
42
F
X
Y
0
1
)
t
0-Y
Now, for each generator y i E F , BT(yi) E Y and because a exists xi E X 6 Q .z)
=
xi,
Finally,
BT(yil
with
= cl(xi).
Thus if we define
is surjective there
6
:
F-
X by
then the outer portion of the diagram is commutative, that is, a 6 = 6 T .
E :
V+
F
so that
6E :
V+
X and
Thus the lower portion of the diagram is commutative, and by definition we
conclude that V
is projective.
8 . 2 . COROLLARY.
Let V
be a finitely generated R-module.
projective if and only if
V
Then V
is
is a direct summand of a free R-module of a finite
rank. Proof.
Let v ,v , . . . , I ) 1
module freely generated by surjective homomorphism F-+
.
direct summand of F . result follows.
be a generating set for V
2
x ,x 1
2
V.
,...,xn. Thus if
and let F
Then the map xi* V
vi
i s projective, then
be a free
determines a V
is a
The converse being true by virtue of Proposition 8.1, the
If we reverse the direction of all the arrows in the definition of a projective module, we then obtain the definition of an injective module. module
V
Thus a given R-
is i n j e c t i v e , if for any given diagram with exact bottom row
43
PROJECl'LV, INJECTIVE AND FLAT MODULES
F\ B
\ \y
\ Y-0
a
there exists an R-homomorphism Y 8 . 3 . PROPOSITION.
(i) An R-module
B = ya.
such that
The following conditions are equivalent: V
is injective
(ii) Any exact sequence 0
+
-& Y+
V -%X
0
of R-modules splits.
Because a is injective, we have the following
Proof. (i) ;+ (ii): commutative daigram:
V
I
0-v
(]I
\ *X
Y is a splitting homomorphism and (ii) follows
Hence Cii)
\ A X
*
(i):
Consider a commutative diagram with exact bottom row
v
I* \
\
\r \
0-v
B
01
\
X
We may harmlessly assume that a is the inclusion map, in which case Y direct sumnand of X, by hypothesis. by
y(y+z) = @(y),y E V,z E 2.
Write X = Y e 2
Then y
and define Y : X-
is an R-homomorphism and
B = ya,
required. 8.4. PROPOSITION.
(i)
@
Let
( V i ) , i E I be a family of R-modules.
Vi. is projective if and only if each Vi is projective
&I (ii)
nVi
*I
is injective if and only if each Vi
is a
is injective.
Then
v as
CHAPTER I
44
Proof.
V
(i) If
0 Vi,
=
GI if and only f so is each Vi. and only if each Vi (ii) Let V
V
is a direct summand of a free module
Hence, by Proposition 8.1, V
and let
V
:
7Ti
---t
Vi and
pi
:
Vi
4
projections and injections, respectively. :
V
-+
Bi
:
Vi
X+
Then
Proposition 8 . 3 ,
V is injective.
Conversely, assume that V
:
X
-+
ni =
is injective and let
Then there exists an R-homomorphism
Vi by
:
there
X+
V by
and so, by
= lv
c1 :
Vi-+ X be an injective
B
X
:
Then ycc = ni($a) = nipi
y = "Ti$.
is injective and
Define $
BiC1.
B is an R-homomorphism with @
$(z) = (Bi(z)1 .
R-homomorphism.
with
vi
that each
Then, for each i E I ,
X be an injective R-homomorphism.
exists an R-homomorphism
Define y
Assume
be the natural
V
iEI
let c1
is projective i
is projective.
nVi
=
then
+
V
vi
with
= 1
=
Ba.
and hence, by
'i Proposition 8.3,
V;
is injective.
Given a pair of rings R,S
we say that V
is an
(R,S)-bimoduZe if
is a left R-module and a right S-module, with the actions of
V
R and S on V
commuting
Given a right R-module
V and a left R-module W,
called the tensor product of free &nodule with
V
x W
V
and
the abelian group
W, is defined as follows.
as a basis; then each element of F
V @ W, R
Let F
can be uniquely
written in the form
with finitely many
zij
distinct from 0.
generated by all elements of the form
be a
Let ,T be the subgroup of F
45
PR05T,CTWErINWCTIVE AND FIAT MODULES
U,Ui E V
and #,Wi E
w.
Then
v8W
is defined as the factor group F/T.
The
R (U,W) under the natural homomorphism
image of
Observe that the z-module V 8 W R (Ui E
v,wi
E
F+
F/T
is denoted by
consists of all finite sums
U
8 #.
I U 8~ w,:
W).
Suppose that V
is an (S,R)-bimodule.
V 8 W can be regarded as a R
Then
(left) S-module by putting
S(U
modules.
,...,Wn
V be an (S,R)-bimodule, and let W
Let
V
v
and
(respectively,
-+
S)
R be left R-
W
as S-modules R R are two free modules over a commutative ring R is an R-basis of
(respectively,W),
V
(V €3 W ) is an R-basis of V 8 W. A ! J R Let R be a ring, V,V' two right R-modules, W,W'
v
w,s E
v o w1 8 ... CB V B wn
ib (wl@ ...@wn)
R Furthermore, if
4:
E
as S-modules
R
(UA)
v,w
It is an immediate consequence of the definition that:
(i) V Q R (ii)
(U E
V and W are modules over a commutative ring R, then V 8 W
In particular, if is an R-module.
8 W ) = su 8 w
V',-$
:
W-+
w'
two R-homomorphisms.
and
then
two left R-modules, and
Then the map
defined by
is a homomorphism of additive groups. bimodules and Let R
Furthermore, if
v
and
V'
are (S,R) -
4 is an S-homomorphism, then 4 8 $ is also an S-homomorphism.
be a ring and let V
be a right R-module.
to be f l a t if for every injective homomorphism f
:
the homomorphism lv@f:
V@W'+V@W R
R
W'-+
The module
W
V is said
of left R-modules,
46
CHAPTER I
It is an easy consequence of the definition that, given a family
is injective. (Vi)
,i E I
of right R-modules, the module
,@ Vi
is flat if and only if each Vi
-LEI
is flat.
The following simple observation will be useful for our subsequent
investigations. 8.5. PROPOSITION.
Proof.
Every projective module is flat.
Since projective modules are isomorphic to direct summands of free
modules (Proposition 8-11, we need only show that the regular module Let f
:
PIr---+ W
be an injective homomorphism of left R-modules.
p r : R @ W ' 4 W',r @ o r++ PW',
and ~-r
:
R
R
9 W+
W,r @
PW
W H
RR is flat.
The maps are R-iso-
morphisms which render commutative the following diagram
is injective and RR
Thus l @ f
.
is flat.
9. PROJECTIVE COWFS.
In what follows all modules are assumed to be finitely generated left modules. Let R be a ring.
We know that each R-module is a homomorphic image of a
projective module.
For some modules
of
projective cover of V epimorphism P-+
only if Kerf (ii) If
V ,V 1
(iii) If Pi cover of
V , f(V')
# f(V).
in case P
V1 @
We say that an R-module P
is a
is projective and there is an essential
V.
ti) A homomorphism f
9.1. LEMMA.
an even stronger assertion is possible.
W of R-modules is said to be essential if for every
A homomorphism f : V+
proper submodule V'
v
:
V+
W of R-modules is essential if and
is superfluous. 2
,...,V
are superfluous submodules of 'L
is a projective cover of
... @ vn.
Vi,
then P @
,
then so is V +..+I'
... @ Pn
n
is a projective
PROJECTIVE COVERS
47
(iv) A projective cover of an irreducible module is indecomposable. Proof. (i) Assume that f V
a submodule of
Vf
If Kerf
is a submodule of
V.
Assume that IJ
Then
V1 + (Vp +
2
+
... +
with f(V) = f(V'),
n
(iii) If fi
:
=
then
V f and so f
=
V
f(W) and so W cannot
V = V'
is superfluous.
+
Kerf
.
(V1 +
... +
V
n
)
V;
cible.
qi: @pi-@Vi
=
=
that @Pi
Hence,
is a projective cover of @Vi.
P-+ V is an essential epimorphism, where V is irreduand Pr',
0 since 0 is the only proper submodule of V .
.
+(P') + +(P")= 0, a contradiction.
required.
V.
is
is superfluous, by virtue of (ii)
If P = P' @ PI', for some proper submodules P'
+(PI) = + ( P f ' ) V
:
=
.
@ Kerfi i=1
qi is essential, which means )I
w
is superfluous, we have
is an essential epimorphism, then
an epimorphism whose kernel
(iv) Assume that
Since
+
n by (i),
is
is essential.
such that
and, since V 1
is
The desired assertion now follows by induction on n.
= V.
Pi+
V
is a submodule of
... + Vn + W) = V
+W
V
then f ( V )
Conversely, assume that Kerf
is superfluous, we must have
Cii)
V
V
W + Kerf,
If W
This proves that V = W and hence that Kerf
V.
a superfluous submodule of
W is an essential homomorphism.
V-+
V =
such that
be a proper submodule of
:
Hence, P
then But then
is indecomposable, as
One of the consequences of the following lemma is that if a module does have a projective cover, then it is unique up to isomorphism. 9 . 2 . LEMMA.
be an R-module such that (i) If M (ii)
+
:
P-+ V .
has a projective cover P, and let M
is a homomorphic image of M. is isomorphic to a direct summand of
has a projective cover L,
M
then P is isomorphic to a direct
L
Ciii) If M Proof.
V
is projective, then P
If M
summand of
V
Assume that an R-module
V,
is another projective cover of
(il Fix an epimorphism Since M
4
:
M+
V
then M Z P. and an essential epimorphism
is projective, there is a commutative diagram
CHAPTER I
48
I
0 with exact row and column. 8
is an epimorphism,
Because
$ is an essential epimorphism and
is also an epimorphism.
is a splitting homomorphism
f
:
P+
P
But
M, and hence M
$8
is projective, so there
= Imf @ Ker 8.
Since f is
injective, (i) is established.
(ii) Assume that epimorphisrn.
L is a projective cover of M,
Then
@f
:
and let f
:
L
-+
M be an
V is an epimorphism, hence by (i) P is
L -+
isomorphic to a direct summand of L. (iii) Suppose that M choose
is a projective cover of
4 to be essential.
which will follow that
M and let P'
= BCM').
and hence P' # P .
M
V,
and in the diagram above
It will be shown that
= Imf 2 P .
Assume that
is also essential, from
M'
is a proper submodule of
Then
Thus 9
is essential, as we wished to show.
'
We have accumulated all the information necessary to prove the following fundamental result.
The statement of this result uses the fact that if R
artinian, then V / J ( R ) V is completely reducible (Propositions 6 . 2 and 6.28). From now on we shall write P ( V ) 9.3. THEOREM.
ti)
Let R
for a projective cover of
be an artinian ring and let V
v.
be an R-module.
V has a projective cover
(ii) A projective R-module
W is a projective cover of V if and only if
is
49
PROJECTIVE COVERS
W/J(R)W
V/J(R)V
g
n V / J ( R ) V = @ Vi is a decomposition as a i=1, direct sum of irreducible R-modules, then V = @ P ( V i ) is a decomposition as i=1 a direct sum of indecomposable R-modules. Furthermore, V . 1 V if and only if (iii) If V
is projective and if
z
P(Vi,
(iv) If V V/J(R)V
and
W are projective R-modules, then V
(i) Write
submodule of L.
V
in the form L/S,
Given a submodule W
canonical homomorphism.
Now choose W
where S,
of
We claim that W
W if and only if
W'
is a direct summand of L;
to be a submodule of L
is surjective, and let p
L/W
:
is projective and S
f,
:
to be minimal in
L/W+
L/S
a
be the
such that f,
is
is essential and S is artinian. if sustained, it will follow that
is a projective module and hence that L/W Choose
L
let
essential; such a submodule exists, since f,
W ' L
1
W/J(R)W
Proof.
L/W
j
= P ( V3. 1 .
is a projective cover of
V.
minimal among those whose projection L
+
L/W be the projection.
Since L
is
projective, there is a commutative diagram
L
with exact row and column, and the minimality of Let L'
L'
L/L' and the two factors are essential. W
ensures that q(L) = W'.
The projection f
denote the kernel of q.
mality of
W'
--f
L/W
Because
forces W = L' (i.e. that
3
factors into
L/S
L' is contained in W, the mini-
4 is an isomorphism).
It follows that
CHAPTER I
50
L
=
W '33 W', as required. be a projective module, let f
(ii) Let W and let a af
:
V
5 J(R)W
hand, f ( J ( R ) W ) 5 J ( R ) V and thus V / J ( R ) W
V/J(R)V.
2:
Y
be an essential epimorphism
Because a
5 Keraf.
is essential, so is
V/J(RlV.
Let f
:
V-'
On the other
Hence Keraf
J(R)W
=
W is a projective 8-
Conversely, assume that
V / J ( R ) V be the projection
V / J ( R ) V be the homomorphism induced by the isomorphism Since f is an essential epimorphism and since W
V / J ( R )V .
projective, there exists an epimorphism g'
5 Kerg = J ( R I W
Kerg'
V
(Lemma 9.l(i) and Proposition 6.9(i)).
and therefore J(R)W
module such that W/J(R)W
W/J(R)W
W+
V / J ( R ) V be the projection.
--+
and thus Keraf
and g : W-
:
:
W--t
with fg' = g.
V
W.
is a superfluous submodule of
V,
projective cover of
Therefore,
is
Hence
w
is a
as asserted.
(iii) By Propositions 6.9(i) and 6.28, J ( R ) V is a superfluous submodule of Since
V
is projective, it follows that P(V/J(R)V) = V .
V.
Invoking Lemma 9.l(iii)
and (iv), we therefore infer that V = P(V
where each P(V.)
and, since Vi
P(V.1.
V . 2: V and since V . 2 V z j ~j assertion follows.
and
... @ P ( V n ) ,
Then, obviously, P(Vi)/J(R)P(Vii)~PP(V.)/J(R)P(V.) 3 3 3 is irreducible, V i / J ( R ) V i = Vi. It follows from (ii) that l.
V a
e3
is indecomposable.
Assume that P ( V . 1
(iv) If
)
obviously implies that P ( V i )
W then obviously V / J ( R ) V
W/J(R)W.
W are projective R-modules such that V / J ( R ) V
2
P(V.), 3
the required
Conversely, assume that 2
W / J ( R )W.
By the Krull-
Schmidt theorem, we may write V / J ( R ) V = V1 '33 where each Vi,Wi
... @ Vn
is irreducible and
V.
W/J(R)W 2
=
w1 @
... @ Wn
Wi, 1 < i < n.
Then, by (iii), we
have
W as required.
=
P(W1)@
... @ P ( V n , = P ( V l ) @ ... '33 P ( V n ,
[/
51
PROJECTIVE COVERS
Suppose that R
is an artinian ring and let n
R where the
=
>
...
U @ 1
@
1 be such that
n'
U : are principal indecomposable R-modules.
Schmidt Theorem, the
are uniquely determined up to isomorphism and the order
Ui
By Proposition 2.12, the above decomposition determines a
in which they appear. complete set
Owing to the Krull-
...,e
{el,
Except when J(R) = 0
3
of primitive idempotents in R
such that Ui
=
Rei.
the principal indecomposable R-modules form only a small However, it is the class which plays
subclass of all indecomposable R-modules.
a very important role, as can be seen from the following: Let R
9.4. PROPOSITION.
be an artinian ring
(i) The following conditions are equivalent: V
(a)
is a projective cover of an irreducible R-module
(b) V
is a projective indecomposable R-module
(c) V
is a principal indecomposable R-module
(ii) If Re ,...,Re then Re /J(R)e
are all nonisomorphic principal indecomposable R-modules,
rn
,...,Rern/JCR)ern are all nonisomorphic irreducible R-modules.
Proof. (i) The implications (a) * (b) and (c) * (a) follow from Lemma 9.1 (iv) and Theorem 9.3(iii), respectively.
Assume that (b) holds.
fl
Since V
projective, there exists n 2 1
such that
direct sum of n
Now the indecomposable components of Rn
copies of R.
principal indecomposable R-modules and
V
=
V @ V',
where Rn
is
denotes a are
is indecomposable; hence (c) follows
by virtue of the Krull-Schmidt theorem. (ii) By Ci) , there exist irreducible R-modules V Rei
=
PtVi),
1
i
=Z rn.
By Theorem 9.3(iii),
irreducible 8-modules.
. .,Vrn
such that are all nonisomorphic
By Propositions 6.9Li) and 6.36, J(R)ei
fluous submodule of Re
i and
Because Rei/J(R)ei
V1,.
,.. .,Vm
is a super-
thus
is completely reducible and Rei
is indecomposable, Lemma
52
CHAPTER I
implies that R e i / J W ) e i
9.l(iii) 9.3Ciiil ,
Vi
.
is in fact irreducible.
Rei/J(R)ei and the result follows.
Hence, by Theorem
10. ALGEBRAS OVER FIELDS. Throughout this section, A
denotes a finite-dimensional algebra over a field F.
All modules are assumed to be finitely generated.
v
Let
be a vector space over
F, let L be a field extension of F, and form the tensor product VL L.
which is then a vector space over
If
{V
.,V
Ire'
n
1 form a basis of F, then the elements {la,...,la n
1
L 0 V, F
=
is a basis of
v
VL over L.
With the
over
aid of the injective F-homomorphism.
I v
VL
v -1Qv
we shall often identify V with its image in V L .
With this identification,
n each element of
n
A.
E L
and
C
V,L, has a unique representation of the form
' 'E
X.V.
V
if and only if all
xi
E
F.
C AiVi i=1
with
Therefore, in passing from
V to VL we are extending the field of operators from F to L. Suppose now that A
is an F-algebra and let AL = L @A.
Then AL
is an
F L-algebra and the map
I
-
A -AL a
1Qa
is an injective homomorphism of F-algebras.
image in AL
and denoting by
element of AL moreover,
{a,,
. .. ,arn1
Again, identifying an F-basis of A ,
can be uniquely written in the form
rn C Xiai E A
rn C Xiai
A with its
we see that each with
Xi E L;
i=1
if and only if all
Xi
E F.
i=1 If
V is an A-module, then the vector space VL becomes an AL-module under
a module action
(A By identifying V
and
1
@a) [A 69 V )
=
A l A 2 Q av
(aE A,v E V,A ,A E L) 1
2
A with their images in VL and AL, respectively, the
53
A L G E B W OVER FIELDS
action above can be written as
(A Thus the action of
. , u 3.
2
E L)
AL on VL is just the extension of the action of A on V
by L-linearity. Suppose now that B homomorphism write BL
0
:
L 8 BF
is an F-subspace of A. L @
F
for the image of B
is a subring (ideal) of
10.1 LEMMA.
Let L
Then there is an injective L-
A defined by 8CR 8 bl under
If B
8.
=
R 8 b,!t
E L,b E B.
We
is a subring (ideal), then B
L
AL.
be a field extension of F
and let
A be an algebra over
F. ti)
If
A 1 , A 2 f...fA n are F-algebras, then as L-algebras
(ii) For all n
1,
[MncA)I L (iii) If I is an ideal of A,
= MnUL’
as L-algebras
AL/IL
as L-algebras
then
1
Civl
J(AIL C_ JULI
(v) If
V,W are A-modules with V
3 W,
(V/WI (vi) If
h @a1
n
V is an A-module, then J C V I L
then
2
as A -modules
L
VL/WL
5 JCV,)
is obviously an injective homomorphism of L-algebras.
Since both of
t h e m are of the same L-dimension, the assertion follows.
(iii) The map
h @ a + A @ (a+ll
*
A @ ( a ,.) ( h 8 a . ,) , A E L,uii E A. 23 23 is an L-algebra homomorphism of AL onto
(ii) The required isomorphism is given by
54
CHAPTER I
(A/IIL
whose kernel contains I L .
fact that the algebras
The desired conclusion now follows from the
(A/IIL and AL/IL
(iv) By Proposition 6.22,
J(A)
are of the same L-dimension.
is nilpotent, hence so is J(A)L.
Now apply
Corollary 6.21. (v) The proof is similar to that of (iii) (vi) Applying (iv), we have
as required.
8
10.2. LEMMA. Let V,W be A-modules and let L
be a field extension of F .
Then Hom(V ,W 1
=
AL L L If V
=
W,
A
the two sides of this are isomorphic L-algebras. Given f E Hom(V,W), we have
proof.
as L-spaces
(Hm(V,W)IL
1@
f E Hom(vL,WL).
A
is an injective L-homomorphism. element
in Ho;(VL,WL)
Hence the map
A
It therefore suffices to verify that each
is of the form $ = CX.(l @ f i )
with
i z hi
E
L,fi
E
Hom(V,Wf.
A Let
{xili
I} be an F-basis of L .
E
uniquely written in the form distinct from 0 .
1 hi
8 V i
is1 Define f i : V+ J,(l @
Then obviously each
Then each element with V i E V
3
of
VL can be
and with finitely many
W by V) =
C Xi iEI
Q fi(v)
fi is an A-homomorphism such that J,
(tJ E V )
=
CA.(l@f.). z
8
i z AII
A-module
L of F, VL
V
Vi
is said to be aberoZuteZy irreducibZe if for any field extension
is an irreducible AL-module.
55
ALC3EBWS OVER FEELDS
10.3. PROPOSITION.
V be an irreducible A-module.
Let
Then the following
statements are equivalent:
V is absolutely irreducible
(i)
(ii) If L
is the algebraic closure of F ,
then VL
is an irreducible A
L
-
module. (iii) End(V)
=
F,
i.e.
each A-endomorphism of
V
is a left multiplication by
A an element of F . Proof. (i) * (ii) Obvious (ii) * (iii) We may harmlessly assume that V Then AL
a faithful irreducible A -module.
L
algebraically closed field L, so A Since A
L
2
is faithful, in which case VL is a simple algebra over an
Mn ( L ) for some n > 1 (Proposition6.27). M (D) for some division algebra
is a simple algebra, we also have A
D over F
and some k
>
is
k
Invoking Lemma lO.l(ii), we deduce that
1.
M n (L)
Mk(L 8 D ) . By looking at the centers of both algebras, we have LZ'LQD. F F Hence D = F and, since End(V) 1Do (Propositions 5.6 and 5.8), we conclude that
A End(l') = F .
A (iii) * (i): Again, we may assume that and hence AL
EZ Mn(L),
V is faithful in which case A
for any field extension L of F ,
a minimal left ideal of M n ( F ) ,
we see that VL
Mn ( F )
Identifying V
with
is a left ideal of M ( L ) with
'
n
= n. Hence VL is irreducible and the result follows. LL Let A be an algebra over a field F . Then F is called a splitting field
dimV
for A
if every irreducible A-module is absolutely irreducible.
10.4. COROLLARY.
F
is a splitting field for A
if and only if
for some positive integers 1' n 2 r " ' r n r *
Proof. Let Vl,...,V
D. = End(Vi)
'
A
21
, 1 < i < r.
be all nonisomorphic irreducible A-modules and let By Propositions 5 . 6 and 5.8,
some positive integers n for A
A/J(A)
By Proposition 10.3,
if and only if Di = D?
z
F
for all
F
i E {l,...,~}.
nrMni
(0:) for i=1 is a splitting field
So the corollary is
CHAPTER I
56
true. Let an ideal I
10.5. PROPOSITION.
Then I
potent elements.
of A
possess a basis consisting of nil-
is nilpotent.
Proof. We may harmlessly assume that F is algebraically closed. more, since
( I + J ( A ) ) / J ( A ) is an ideal of A / J ( A )
having a basis consisting of Thus A
nilpotent elements, we may assume that J ( A ) = 0.
Further-
=
nrMni (F)
for some
i=1
positive integers n ,.-.,n r
M
“i iE
(F), 1 g 11,.
i g r.
..,r}.
i the projection of I
71
It clearly suffices to verify that 7T.U) = 0 z
Since T i ( T )
a . U ) = Mn ( F ) .
and we denote by
is an ideal of Mn (F), either
for all
(I)= 0
or
In the latter case it will follow that Mn (F) has a basis
i
i consisting of nilpotent elements.
10.6. PROPOSITION.
Let A
Thus
= 0
as required.
Thus z E I C_ J ( A )
(F)
i
Then J ( Z ( A ) ) = z ( A ) n J ( A ) .
is nilpotent and hence I = Az
is a
.
and therefore J ( Z ( A ) c - Z(A) n J M ) . is nilpotent and hence z E J ( Z ( A ) ) .
Conversely, if z E ZU) n J(AI, then z 10.7. PROPOSITION.
T.(I)
be an F-algebra.
If z E J ( Z U ) ) , then z
nilpotent ideal of A .
.
But then the trace of each matrix in M,
would be zero, which is impossible.
Proof.
Ti
i
into
If F is a splitting field for A ,
then F
is a splitting
field for Z ( A ) . Proof.
By Proposition 10.6, we have ZU)/J(Z(A))
C Z Y l ) + J(A))/J(A) 5 z ( A / J ( A ) )
Invoking Corollary 10.4, we deduce that Z ( A ) / J ( Z ( A ) ) is a subalgebra of a direct product of finitely many copies of F.
But any such subalgebra must also be a
.
direct product of finitely many copies of F.
of Corollary 10.4.
Hence the result follows by virtue
We now turn our attention to direct decomposition of algebras.
First, however,
we must develop our vocabulary. By an F-representation of an F-algebra A we understand any homomorphism A+
EndW
F
(or simply a representation of A )
of F-algebras, where V is a (finite
WEB-
57
OVER FIELDS
If n = dimV, then End(V)
-dimensional) vector space over F.
F with the F-algebra Mn(F). mrphian A+
is identifiable
F
Thus the given representation of A
Mn (F) J we shall refer to any
defines a hano-
such homomorphism as a mutfix
Pepesentation of A . If f : A+
is a representation of A ,
End(V)
define xu
=
f( x ) V for
F x E A,V E
of f.
V.
V becomes an A-module, called the underzying moduZe
In this way
Conversely, if
V is an A-module, define f
..
:A
EndCV) by f( x ) v = xu
F
V.
for x E A,v E
Then
f is a representation of A ;
in case V is the regular
A-module, we shall refer to f as the regular representation of A. Two representations
fd
: A
--C
there exists an F-isomorphism JI :
End(Vi)
,i
F Vl+ V
f b) =
= 1,2, are said to be
such that
w1(3)JI-l
It is easily seen that two representations of A the underlying modules are isomorphic.
f
:
A
+
equivalent if
for all x
E A
are equivalent if and only if
We say that a representation
is CrreQcible (completely reducible, indecomposable) if the
End(V)
F underlying module
V is irreducible Ccompletely reducible, indecomposable); the
same terminology will be applied to matrix representations.
10.8. PROPOGITION.
Let
A be an P-algebra.
(i) There exists a direct decomposition
of A
into indecomposable two-sided ideals Bi # 0 with B . B .
(ii) Write
1 =e
i
... + en
2 3
with
ei E Bi.
Then the ei
=
0 for i # j
are mutually
orthogonal centrally primitive idempotents and Bi = Aei = e$. (iii) Z ( A )
=
ZWe
1
... @ Z(A)en
indecomposable ideals and
is a direct decomposition of Z U )
into
Z(A)ei = Bi n Z U ) .
(iv) J(Z(A)eil = J ( , Z ( A ) ) s i and Z ( A ) s . / J C Z ( A ) e . ) is a finite field extension of 2
2
F. (v) Let Pi
=
Z(Alei/J(Z ( A l e i ) , let Bi
:
Z(A) + Pi be the natural homomorphism
CWTER I
58
and let fi ' i: X
yi
Mn
: Z(A)--t
be the regular representation of Fi.
Mni(F) (F)
by
yi
=
fiei.
Define
Then
i
{yl,.
.. ,y 1
tation of
Z(A).
and
is a complete set of nonequivalent irreducible matrix represenEach yi
satisfies y . ( e . ) z
=
z
y . ( e . ) = 0 for j # i. 2 3 then each ni = 1.
1 and
Furthermore, if F is a splitting field for Z ( A ) Proof.
(i) and (ii).
Apply Propositions I.l(iii) and 4.3
(iii) The first statement and the inclusion Z ( A ) e . z Suppose that Z E B . n Z ( A ) .
Since e
2
have z = ze
i
E Z(A)ei,
n Z(A)
follows from (ii).
i is the identity element of Bi, we
proving that Bi n Z ( A )
(iv) By Proposition 6.35,
5 B z.
5 Z ( A) e i .
J(Z(A)ei) = J ( Z U ) ) e i .
Because each ei
is
primitive, Proposition 9.4(ii) ensures that the Z (/!)-module Z ( A ) e i / J ( Z ( A ) e i ) irreducible.
is
The latter implies that the commutative algebra Z(A)ei/J(Z(A)ei)
is simple and hence is a field.
That this field is a finite extension of
follows from the assumptions that A
F
is finite-dimensional over F .
(v) It is obvious that Keryi = Kere;.
Assume t h a t
Z
E Kerei.
Then Z e i =ze i
for some x E J ( Z ( A ) ) , so
and therefore Kere. c Z ( A ) (l-ei) 2 -
obvious, we conclude that Kere
i
+ =
J(Z(A))ei.
The opposite inclusion being
ZU)(l-eil + J ( Z ( A ) ) e i .
It follows from (iii) and (ivl that
and that each Fi Z(A),
is a field.
then each Fi
Furthermore, if F
and hence ni = 1.
is isomorphic to F
only irreducible representation of P
if
is a splitting field for
the result follows.
We now introduce the important concept of blocks.
n A = .@B 2=1
n
i
=
@Ae
i=1
i
Let
Since fi is the
ALGEBRAS OVER FIELDS
be the decomposition defined in Proposition 10.8.
59
We shall refer to Bi
block and to ei
as a block idempotent of A.
indicate that B
is a block containing the block idempotent e.
as a
B = B(e) to
We shall also write
The represen-
--+
(F) described in Proposition 10.8 is uniquely determined (A) Mni up to equivalence by ei. We call y the irreducible representation of Z U ) tation Y i
:
i
associated with
e
i'
By a central character of A Z(A) 3 F.
y l , ...,yn
If F
we understand any F-algebra homomorphism
is a splitting field for
are all central characters of A;
central character of A
associated with
10.9. PROPOSITION.
Let F
number of blocks of
A.
ei
Z(A), then by Proposition 10.8, we shall refer to yi (or Bi).
be a splitting field for Z(A)
and let n
be the
Then dimJ(Z(A)) = dimZU)
F
F
-
n
As we have seen in the proof of Proposition 10.8(v),
Proof.
as the
copies of F.
is isomorphic to a direct sum of n
Z(A)/J(Z (A))
Hence
and the result follows. 10.10. PROPOSITION.
A
Let A = B(e
1
)
@
... 8 B(en )
be the direct decomposition of
into sum of blocks.
n (i) For every A-module modules.
i E C1,2,.
V, V
In particular, if
8 ei V is a direct decomposition of 'L i=1
=
V
is indecomposable, then V = e . V
into A-
for a unique
.. ,nl
(ii) For each left ideal I of A ,
we have
n
I = i=1 8 (I n B(ei)) In particular, each indecomposable left ideal lies in exactly one of the B ( e i ) . (iii) If
A
and A 2
are ideals of
are direct sums of blocks.
A and A
=
A 1 @ A 2 , then A
and A
In particular, the blocks are the only indecomposable
CHAPTER 1
60
ideals of A . (iv) Let I be an ideal of
of
Z(A)
associated with e
Proof.
Z(A)
Then ei E I
i'
(i) Because ei
and let yi
be the irreducible representation if and only if y i ( l ) # 0.
is central, e . V is an A-module. 2
Since the e i
are orthogonal, we also have
n
n
C e.V = @ e i V i=l
Taking into account that
n
n
the assertion is proved.
n (ii) By (i), I = @ e . A and so i=1
Hence
n
n
Hence for all i
and since B ( e .) 2
is indecomposable we have B ( e i )
5A l
or B ( e i )
5A l .
Thus
as asserted.
(iv) If ei E I then y i ( e i ) = 1 # 0 that yi (1)# 0. have
Because y i ( I )
and hence y . ( 1 ) # 0 . 2
Conversely, assume
is a nonzero ideal of the field y i ( Z ( A ) )
,
we
ALGEBRAS OVER FIELDS
61
It therefore follows,from Proposition 10.8(v), that there
Thus y i ( e i ) E y i ( I ) .
are elements x E I , y E J ( Z ( A ))
x Multiplying both sides by
e
E Z(A)
and
=
+
ei
z(1-e
i yields ei
such that
+y
.)
= xe i - y e i .
Hence, by raising both
sides to the k-th power, we find
ei with u
I.
Since yei
=
u
+
k
k
(-1) ( y e i )
is nilpotent, the result follows.
Let e ,e , . . . , e n be the block idempotents of A 1
i E I1.2,
...,n}
such that
2
B(ei),
and e . V = 0 3
.
Observe that if
V
lies
then
(a) A l l A-modules isomorphic to V t, = 1-t, =
j # i
for all
l i e s i n the block B ( e i )
In this case we say that V
(b)
be an
It follows from Proposition 10.10 that there exists
e.V = V
in the block
and let V
2
indecomposable A-module.
.
lie in B(ei)
for all v E V
e.2,
The above gives a classification of all indecomposable (and in particular, all irreducible) A-modules into blocks. principal indecomposable modules.
Our next aim is to tie together blocks and
The following preliminary observations will
clear our path. Let e
10.11. LEMMA.
be an idempotent of A.
Then
n I = Ie
(i) For any ideal I of A ,
Ae
(ii) Ae/J(A)e
denotes the image of
Proof.
where
-e
(i) Suppose that x E Ae
n I. Then x
Therefore
x = ae = ae2 = xe E Ie
e
in = ae
A'
= A/J(A).
for some a E A .
CHAPTER I
62
proving that Ae n I
5 re.
The opposite containment being trivial, (i) follows
JU), we have
(ii) Applying (i) for I =
.
as required.
Note that if Ae
is a principal indecomposable A-module, then by Proposition
2;
9.4(ii) and Lemma lO.ll(ii),
is an irreducible A-module.
n
10.12. LEMMA.
Let A =
@
composable A-modules. for any A-module
ei
Let
denote the image of
ei
into principal inde-
=
dim HomUe
F
F Z
Then
--
V)
=
rndim End(A e J
F
A
factor of V . By Lemma 5.5, the map Hom(Aei ,v)
A morphism of additive groups;
-
;
A
is the multiplicity of the irreducible A-module
Proof.
1 = A/J(A).
in
V, dime .V
where m
be a decomposition of A
Aei
i=1
jei
as a composition
eiV, f++ f ( e )
is an iso-
Since this map is obviously F-linear, we have
e .V
Hom(Aei,V)
as F-spaces
A The equality dimHomUe
F
V)
= rndim End(2;)
Therefore we may assume that Aei
V is irreducible.
is trivial if
F
i'
A
-
V has a maximal nonzero submodule X. 4 X+
is projective, the exact sequence 0
V - + V/X
0
Because
induces an exact
sequence 0
3
Homue
A
i'
X) --t Hom(Aei,V) 4 Hom(Ae A i' V/X) A
+
0
Thus w e have dim Hom(Ae
F If m, and
and rn
V/X,
A
i'
V) = dim HomUe
F
A
are the multiplicities of
''
X) + dim HomUe V/X) F A
"
zei
as a composition factor of
respectively, then applying induction on dim V ,
we have
X
ALGEBRAS OVER FIELDS
63
dim Hom(f!ei,X) = rn dim EndgZd
F
' F A
A
dim Hom(Aei,V/X) = rn dim End(zii) F A ' F A Adding up these equalities gives dim Hom(Aei,V)
F
as asserted. As
" F A =
.
rndim End (2eil
F
A
an immediate consequence, we derive
10.13. COROLLARY. of e
+m )dim End(Azi)
= (m
A
in
=
Let e be a primitive idempotent of A,
A/J(A)
and let V be an A-module.
e
let
be the image
Then the following conditions
are equivalent: (i)
1;
v
is a composition factor of
(ii) Hom(Ae,V) # 0
A (iiil
eV # 0.
9
U and V be principal indecomposable A-modules.
Let
V are linked, written U
- V,
We say that U
if there exists a sequence U
=
U,.. .,Un
=
and
V of
principal indecomposable modules such that any two neighbouring left ideals in this sequence have a common composition factor.
It is clear that
-
is an
equivalence relation on the set of all principal indecomposable A-modules what follows we denote by X 1
10.14. PROPOSITION.
Let
A
,...,Xrn n = @ Aei
the equivalence classes of be a decomposition of A
i=1 indecomposable modules, and for each jE 11,.
..,rn}
.
In
-. into principal
put
I . = @Ae
,,EXj ti)
...,Irn
11,
are all blocks of
A
(ii) Two principal indecomposable A-modules are linked if and only if they.belong to the same block.
64
CHAPTER I
rn (i) It is obvious that A = 8 I j=l j '
Proof.
suffices to show that each I
j
By Proposition l.lO(iii),
it
is a two-sided ideal contained in a block.
Assume that Ae. c I. and Aek C I k with j # k . Then Aei and Aek have no 2 3 composition factor in common. In particular, Aek has no composition factor Thus, by Corollary 10.13, ez.Aek = 0.
isomorphic to
I.I = 0 for j # k . 3 1 1
This proves that
Hence
rn I A = I.( 8 Ik)= I T c I . 3 3 g=1 3 - 3
and so I
is a two-sided ideal of A.
i
If Aei
--
and Ae
j
have the composition factor Aek in common, then by
Corollary 10.13, ekAei # 0 and exist blocks B,B',B" that 0 # ekAei
E"B
e Ae # 0. By Proposition lO.lO(ii),
k j
such that Aei
5B ,
and hence B
B".
=
Ae. 3 -c B'
and Aek -c E r r .
there
It follows
Similarly, 0 # e Ae . c B"B' k 3 -
and so
Consequently Ae.+Ae. C B and repeated application of this argument -L 3 This establishes the shows that I i s contained in a certain block of A. j required assertion
B' = B".
(ii) Direct consequence of (i). Assume that F is a splitting field for A module.
Then End(V1 = F
and let V
and each element z E ZU)
be an irreducible A-
acts as an element of
A End(V) on V .
Thus there is a function $ : z(A)---+F
such that
A
Clearly J,
is a central character of A.
A afforded by
character of
as the central
V.
Suppose that F
10.15. PROPOSITION.
We shall refer to J,
is a splitting field for
A.
Then two
irreducible A-modules belong to the same block if and only if they afford the same central characters. Proof.
Assume that the irreducible A-modules
characters p
and
JI,
respectively.
If U
and
U and V afford the central
V belong to a block B
=
BCe)
ALGEBRAS OVER FIELDS
then eu = u
and eu
for all u
= U
U,U E V .
E
65
proving that p = $ by virtue of Proposition 10.8(v). p = $,
belong to B ( e ) .
and let U
eu = $(e)u = U
for all U E V .
p(e) = @ ( e )= 1,
Hence
Then P(e)
=
@ ( e )= 1
V
This shows that
Conversely, suppose that and hence
lies in B
and the result
follows. Let e 1
,...,en
be a complete set of primitive idempotents of
A
5
Ae @
Then
... @Aen
A as a direct sum of principal indecomposable modules.
is a decomposition of Let
=
A.
e
denote the image of
i in 2
= A/J(A).
We know, from Lemma lO.ll(ii),
that Aei/J A ei
2
--
A ei
Suppose that the numbering is so chosen that Ae
nonisomorphic irreducible A-modules,
ij
C
C =
(c..) ZJ
j
-_
are all
..,m},
let ‘ij be the as a composition factor of Aei.
are called the Cartan invariants of A,
The nonnegative integers c
m matrix
2;
are all nonisomorphic
2; ,...,Aern
Given i , j E 11,.
multiplicity of the irreducible A-module
x
,...,Aern
Then, by Proposition 9.4,
indecomposable A-modules.
rn
1
is called the Cartan matrix of
A.
and the
The Cartan matrix
is only determined up to a permutation of its rows and columns (depending on
the numbering of Ae 10.16. PROPOSITION.
,. ..,Aem ) . Let F
be a splitting field for A,
A be a decomposition of A
=
Ae @
let
... @Aen
into direct sum of principal indecomposable modules,
and let C = ( C i j ) be the Cartan matrix of (i) cij = dime& F J (ii) The multiplicity of Aei Krull-Schmidt theorem is dim1 F (iii) The multiplicity of
A.
Then
as a direct summand of
A in the sense of the
ei as a composition factor of A
is dimezA F
CHAPTER X
66
proof.
(i) Since F
is a splitting field for A, End(jei) = F .
Lemma 10.12, we deduce that the multiplicity of
Aei is equal to dime l e 3 i'
z ei
Applying
A
as a composition factor of
as required.
$9
(ii) Since F
is a splitting field for A ,
rings over F .
Hence the multiplicity of
Owing to Proposition 9 . 4 ,
Aei
2; i
is a direct sum of full matrix in
is equal to
is determined, up to isomorphism, by Aei
therefore a i d e i also equals the multiplicity of Aei F (iii) Direct consequence of Lemma 10.12. m As
- -F
. and
as a direct sunnnand of
A
an inmediate consequence, we derive
10.17. COROLLARY.
Let F
be a splitting field for A,
nonisomorphic irreducible A-modules and let ni = dimV
if
r
In particular,
r
let V1,...,V 1
i < r.
r be all Then
.
67
2 Group algebras and their modules d i s c u s s a number of t h e i r proper-
I n t h i s chapter w e introduce group a l g e b r a s an t i e s t o be a p p l i e d i n subsequent i n v e s t i g a t i o n s .
These include support of
c e n t r a l idempotents, t h e notion of d e f e c t group of a block, t h e number of irred u c i b l e FG-modules and some formal p r o p e r t i e s of induced modules.
Later c h a p t e r s
w i l l t r e a t v a r i o u s aspects of t h e s e t o p i c s i n g r e a t e r d e t a i l and depth. s e l e c t i n g m a t e r i a l w e have held s t e a d f a s t t o a single-minded purpose:
In t o present
o n l y those r e s u l t s deemed e s s e n t i a l f o r t h e a p p l i c a t i o n t o t h e study of t h e Jacobson r a d i c a l of t h e group algebra
FG.
1. GROUP ALGEBRAS
R
Let
group a l g e b r a
RG
induced by t h a t i n combinations
(ii) Cx * g
9
(iv)
+
be a group, p o s s i b l y i n f i n i t e .
i s t h e f r e e R-module on t h e elements of G.
More e x p l i c i t e l y ,
Cx*g,x
9 ( i ) Cx*g =Cy * g g g
(iii)
G
be a commutative r i n g and l e t
E R,g E G , 9 i f and only i f
RG
9
-
yg
Z
=
for a l l
x # 0 subject t o
9 g E G
Cy * g = CCx +y 1 * g 9 g g
(Ex * g ) (Cyh*h)
=
Czt*t where
9 r(Cx * g ) = C k x ) * g 9 9
for a l l
Cxgyh gh=t
r E R
It i s straightforward t o v e r i f y t h a t these operations define
R-algebra with G,
withmultiplication
c o n s i s t s of a l l formal l i n e a r
with f i n i t e l y many
x -
G,
The
1 = lR*lG where
respectively.
lR and
lG
RG
as anassociative
a r e i d e n t i t y elements of
R
and
With t h e a i d of t h e i n j e c t i v e homomorphisms
we s h a l l i n t h e f u t u r e i d e n t i f y
R
and
G
with t h e i r images i n
RG.
With t h e s e
i d e n t i f i c a t i o n s , the formal sums and products become o r d i n a r y sums and products
CHAPTER I1
68
We shall also adopt the
For this reason,from now on we drop the dot in xg.g. convention that RG Let x = Cz
2
RH means an isomorphism of R-algebras. Then the support of
E RG.
x, Supp, is defined by
It i s clear that Suppx is a finite subset of G 3: =
0.
that i s empty if and only if
We next exhibit some elementary properties of group algebras.
1.1. PROPOSITION.
Let A
be a homomorphism of
f : RG 3 A
G
be an R-algebra and let
into the unit group of A .
Then the mapping
defined by
i s a homomorphism of R-algebras.
In particular, if
R-free with JIG) as a basis, then RG Proof. Because RG of R-modules.
JI is injective and
A
is
A.
is R-free with G as a basis, f i s a homomorphism
Let
x be two elements of RG.
=
Ex
g
g
and y = Cy
s9
Then
as asserted. The augmentation ideal homomorphism from RG
to
I(R,G) consists of all x
I(R,G)
of RG i s defined to be the kernel of the
R induced by collapsing =
ZX
99 E RG
for which
G
to
1.
In other words,
69
QROUP ALGEBRAS
as the augmentation of x.
We shall refer to aug(z)
It follows from the equality CXc$
= CXgCg-l)
+ Cx
is a free module with the elements g-1 (1 # g
that as an R-module, I(R,G)
In the future we shall often suppress reference to R
as a basis.
denote the augmentation ideal of RG by
ideals of RG, respectively, generated by RG*X = C RGx and
and simply
and X*RG for the left and write
X*RG = C xRG ~X
for all x E X, then RG*X = X*RG is a two sided ideal
Of course, if RGx = xRG
For example, if N (n-llg
G)
X, i.e.
6 X
of RG.
E
I(G).
is a subset of RG, we write RG‘X
If X
9
=
g(g
show that RG*IW) = I(N)*RG
G, then the equalities
is a normal subgroup of
-1 ng-1)
and g(n-1) = (gng-l-l)g
is a two-sided ideal of RG.
( n E N,g
E
G)
The signigicance of
this ideal comes from the following fact.
1.2. PROPOSITION.
N
= Kerq.
Let J, : G+
H be a surjective homomorphism of groups and let
Then the mapping f : RG-
RH which is the R-linear extension of J,
is a surjective homomorphism of R-algebras whose kernel is RG*I(N).
In
particular,
Proof.
That f is a surjective homomorphism of R-algebras is a consequence
of Proposition 1.1.
It is plain that RG*I(N) 5 Kerf.
induces a homomorphism
is an isomorphism.
7
:
RG/RG*I(Nl--t RH
.
Thanks to Proposition 1.1,
Consequently, f
The restriction of
I-’
7,
can be extended to a homo-
CHAPTER I1 70
RG/RG*I(N)
morphism RH+
7.
which is the inverse to
Thus Kerf = RG.Z(N),
as desired. Let I be an ideal of RG and let
Then G n (l+I) is the multiplicative kernel of the natural map Ghence a normal subgroup of G.
RG/I,
and
In view of this obsefvation, the next corollary
arises from Proposition 1.2 by taking I = R G * I ( N ) . 1.3. COROLLARY.
Let N
be a normal subgroup of
Let H be a subgroup of G.
G.
Then
Because RH is a subring of RG,
we can view RG
as a left or right RH-module by way of ordinary multiplication. 1.4. PROPOSITION.
Let H be a subgroup of G.
transversal for H
in G, then RG
If T
is a right (left)
is a free left (right) RH-module with T
as
a basis. Proof.
T is a right transversal for H in
Assume that
t E T,IRH)t
G.
Then, for any
is the R-linear span of the coset Ht.
Accordingly, for any n tl,t*,...,tnET,(RH)tl + + (RH)t, is the R-linear span of UHti As is i=1 apparent from the definition of RG, if X and Y are disjoint subsets of G ,
...
their R-linear spans meet at 0.
.
Hence RG = @ (RH)t, proving that RG
is a
FT free left RH-module with where T
T as a basis.
A similar argument proves the case
is a left transversal.
1.5. PROPOSITION.
Let S be a subring of a commutative ring
group G, R B S G
RG
R.
Then for any
as R-algebras.
S
Proof.
Observe that
11 @ g1g E GI
is an R-basis for R 03 SG and that the S
mapping
GROUP ALGEBRAS
For any groups G
1.6. PROPOSITION.
and
R(GxH)
Proof.
Again note that
.
Now apply Proposition 1.1.
is an injective homomorphism.
{X
2
71
H,
RG 8 RH R
8 y l x , y E G}
is an R-basis for RG 8 R H R
and
that the mapping
is an injective homomorphism.
1.7. PROPOSITION. G
Let
Now apply Proposition 1.1.
{cili E 1) be the set of all finite conjugacy classes'of
.
and, for each i E I, let
all
.
Then ZCRG)
is the R-linear span of
.c; Proof.
i E I.
Since for all g E G; g-'C:g
On the other hand, let x
or, equivalently z
i'
EX
99 E
C'
C Z(RG)
for all
z -
Then, for all t E G , t - ' z t = z
Z W ) .
for all t E G,g E Supp.
= z
t-lgt
=
we have
= C+
Hence x
$7'
for which Ci
linear combination of those 6';
S u p p , as required.
.
is an R-
The "if" part of the following result is a classical Maschke's theorem 1.8. PROPOSITION.
istic p
> 0.
order o f
G.
Proof.
be a finite group and let F
Let G
Then FG
is semisimple if and only if p
If p > 0 divides
and x E ZVG).
IGl,
then z
by
W'
and let 8
:
V-'
W
=
Cg
does not divide the
satisfies 'X
SfG
J V G ) # 0.
Hence FGx
let W be a submodule of an FGmodule F-subspace
be a field of character-
Conversely, assume that p V.
Write
V = W 8 W'
be the projection map.
= 0
=
1 IGI,
and
for a suitable
Define $
:
v--+ 'L
CHAPTER I1
72
+ ( y v ) = 1 ~ 1 c- lXex-lya = I G I - ~c (yz)e(yz)-'yu xEG ZEG 1 G I - l C yzez$7EG
=
$
1
u
= y$(~),
i s an FG-homomorphism. Suppose now t h a t
xex-12,
Accordingly,
W"
U E =
2,
W.
Then, f o r any
and
$(Ul = U.
V
is an FG-submcdule of
by t h e above,
- $(u) E
21
W"
Setting
W" n W
such t h a t
V r r , so
x E G,x-'v
U = $(U)
+
= 0.
E W,
-1
= Ker$,
it follows t h a t
u E V.
Finally, l e t
W+W'.
(V-$(U))E
u.
e(3c-l~) = 5
so
Then,
V = d J @ W"
Thus
and t h e r e s u l t follows. 1.9. PROPOSITION.
Let
normal p'-subgroup of
F
be a f i e l d of c h a r a c t e r i s t i c
G
c
e = 1Nl-l
and l e t
x.
Then
p > 0, l e t N be a e
i s a c e n t r a l idem-
xEN
potent of
FG
f a c t o r of
FG.
Proof.
f
:
Kerf = FG-I(N),
Thus
F(G/N).
In p a r t i c u l a r ,
e
it s u f f i c e s t o show t h a t
we a l s o have
FG*I(N) C - FG(1-e)
ne = e
F(G/N)
is a direct
is a c e n t r a l idempotent.
be t h e n a t u r a l homomorphism.
f ( e ) = 1, w e have 1-e E K e r f
n E N,
Given
2
I t i s easy t o v e r i f y t h a t
FG + F(G/N)
Since
FGe
such t h a t
Since
Let
FG = FGe @ FG(1-e)
and
FG*ICN) = FG(1-e). and t h e r e f o r e
FG(1-e)
(n-1)s = 0 and
so t h a t
5 FG*I(N).
FG-I(N)e = 0.
and t h e r e s u l t follows.
We c l o s e t h i s s e c t i o n by providing t h e following p i e c e of information. Let
R
G
be a commutative r i n g and
G
a r e p r e s e n t a t i o n of
V
on
a group.
V,
Given a nonzero R-module
by
we understand a homomorphism
p : G
-t
Aut Cv'l
R W e say t h a t
p
representation
is faithfd i f P
of
G
on
K e r p = 1.
V,
Owing t o Proposition 1.1, t o each
t h e r e corresponds an RG-module s t r u c t u r e on
V
given by
CCx glv = ZX g
Conversely, any nonzero RG-module
V given by
p @ ) O = gv
for a l l
9
p(g)U
V determines a r e p r e s e n t a t i o n
g E G, U E V .
(U E p
of
v, cc9 E G
Hence t h e r e is a b i j e c t i v e
on
R)
CENTRAL IDEMPOTENTS
I3
correspondence between t h e c l a s s of a l l nonzero RG-modules and t h e c l a s s of a l l r e p r e s e n t a t i o n s of
G
on R-modules.
V
Moreover, i f
i s R-free of f i n i t e rank
2 GL(n,R) , t h e l a t t e r being t h e group of a l l n x n nonR Thus t o each r e p r e s e n t a t i o n of G on t h e s i n g u l a r m a t r i c e s with e n t r i e s i n R.
n , then
AutCV)
R-free R-module p* : G -
V
GL(n,R).
n , t h e r e i s a matrix r e p r e s e n t a t i o n
of f i n i t e rank
R = F
In particular, i f
generated F-modules a r e f r e e of f i n i t e rank.
F
I n t h i s case t h e study of FG-
G on F-modules, and matrix r e p r e s e n t a t i o n s of
modules, r e p r e s e n t a t i o n s of over
i s a f i e l d , then a l l f i n i t e l y
G
a r e e s s e n t i a l l y equivalent.
1.10. PROPOSITION.
Let
be a commutative r i n g .
RG.I(H)
H be a subgroup of an a r b i t r a r y group X
If
i s a generating set f o r
C RG(x-1)
=
and
then
I ( H ) R G = C (x-1)RG XGX
XEX
Proof.
H,
G and l e t R
W e s h a l l only e s t a b l i s h t h e f i r s t e q u a l i t y , s i n c e t h e second follows
by a similar argument.
C RG(x-1) 5 R G * I ( H ) . x EX i n El, h-1 E C RG(x-1).
I t i s clear t h a t
h
only show that f o r any n o n i d e n t i t y
Thus we need
h
Now
is a
XEX
group word i n t h e
xi E X and w e proceed by induction on t h e l e n g t h
F i r s t it i s t r u e f o r words of l e n g t h
word.
xYl-1
x -1 E C RG(x-1)
2
XEX Suppose it i s t r u e f o r words t h e form
*1
x; h and both
h-1
=
1 :
-x.-1 [x;-l)
E
C RG(x-1) XEX
h of l e n g t h and
t of t h e
xi1-
t.
Any word of l e n g t h
1 are in
C RG(x-1). XEX
t+l i s of
Hence
a s desired. 2. CENTRAL IDEMPOTENTS.
Throughout t h i s s e c t i o n , istic
p > 0.
G denotes a f i n i t e group and F a f i e l d of c h a r a c t e r -
Our aim i s t o provide information on support of c e n t r a l idem-
p o t e n t s of t h e group a l g e b r a Let
A
FG.
be an F-algebra and d e f i n e
[A,A],
t h e canmutator subspace of
A , to
74
CHAPTER I1
be the F-linear span of all Lie products
[Z,yI =
x p y x with Z,y
E
A.
The
following observation illustrates how the commutator subspace can be brought into argument.
2.1. LEMMA.
Let A
a 1 ,a 2 ,...,a rn
E A
Proof.
be an algebra over a field F of characteristic p > 0.
and if n
5
0 is a given integer, then
By induction argument, we need only prove that
n (a+blP for a l l
If
a,b E A .
n
2
t
n
Suppose first that
n bp (mod I A , A I )
= 1.
Expanding by the distributive
law,
+ bp +
( a t b ) P = ap
la a 1 2
...
where the sum is over a l l products a equal to a
or b.
a of p terms, ai E { a , b } , not all 1 P With each word a a a associate its cyclic permutations 1
a a 1
2
...ap
,a a 2 3
1
we have I-y
=
...ai
ai ai
,y
=
P
2
[y,bl E [ A , A I
... P p l
1
[A,AI.
This is so since for
...aipai ...a'.j-1
a 2. J.a2. j+l
1
where
...a .
y=a.a 1
2
...a a ,...,aPa ...aP-1
All these products are congruent modulo x =
...aP
and
6 = a.
'j-1
2
a.
' j 'j+l
Therefore the sum of these cyclic permutations is px hence it belongs to (2)
[A,A].
1
...ai
...x P
p
modulo
[A,AI,
and
Invoking (1)I it follows that
(xty)P =
B
t yp t 2
(X E
and hence that
@pySIp
(wIp-(&dP(mod = Is,(y3c)p-1E/1 oaoa
[A,All IA,AI I
[AIAI)
75
CENTRAL IDEMPOTENTS
Accordingly
s E IA,AI
(3)
implies
2’ E
[A,AI
n.
The r e s u l t now follows from (2) and (31 by induction on
The n e x t lemma w i l l enable us t o e x p l o i t t h e previous r e s u l t .
follows
denotes t h e set of all conjugacy c l a s s e s of
CR(G1
Proof.
C = {g
Given
n A . E F and 2
c
h
i=1
. = 0. ’
...,g,,
,
E CRCG) n-1
x
E [FG,FGl.
Since
Cgt-3 t
=
IFG,FGI
Let
H be a subgroup of
G
x
=
- t( g t - l )
( t , g E G)
i s spanned by a l l
zy-p with r , y
E G,
-1
q-yx
t h e r e s u l t follows by v i r t u e of the e q u a l i t y
G.
C Aigi, where i=1 and so i n view of t h e i d e n t i t y
assume t h a t
x = c Ai (gi-g,) i= 1
Then
g-tgt-1 we have
,
I n what
= q - 3 ~ (zycy)~.
Then t h e n a t u r a l p r o j e c t i o n
defined by
“8’‘,EHX$
n( c g EG i s obviously an F-linear map.
define
+ X
I n what f o l l o w s , f o r each s u b s e t
X.
t o be t h e sum of a l l elements i n
E FG
X
G,
of
we
The next r e s u l t , due t o
Brauer , w i l l be of fundamental importance. 2.3. THEOREM. subgroup of IT : F G d
P be a p-subgroup of
Let
G
for which
5 H 5 N(P1. G
Let
and assume t h a t
C ,C 1
2
C’
,...,Cp
Z(FG)
Z(FH)
into
n S
=
0,
be a l l conjugacy classes of
G,
with
Xk # 0 f o r k E
By P r o p o s i t i o n 1 . 7 , t h e elements Note t h a t
let
C E CI(G)
(1
H
S = C(P)
and l e t G Then t h e n a t u r a l p r o j e c t i o n
FS induces a r i n g homomorphism of
t h e F - l i n e a r span of a l l Proof.
S
G,
,...,t}
+ + .: Cl,Cp ,..C
C
and
and
Xk
=
0
for
be a
whose k e r n e l i s
let
Xi
= Ci n S
k E {ttl, ...,r ) .
c o n s t i t u t e a n F-basis for
Z @‘GI
.
CHAPTER I1
I6
,...,Xt
X
and t h a t
a r e mutually d i s j o i n t .
H.
union of conjugacy c l a s s e s of
ZWG)
to
Z(FH)
Hence
1T
8 E
i n which case
Ts
Ts with s
++ n(C.C.)
=
2 3
E S.
i,j
0 =
3
=
=
(z,y)
where
x,y
with
E
and
0,
E
I l l . . .,PI,
xy
s)
=
T
and denote by
X . = 0 or
then e i t h e r
2’ #
Now suppose t h a t
0.
xj
=
0,
Then
P a c t s as a
it follows t h a t
(hsh-l,hyh-l).
Because
ranges a l l o r b i t s of
On t h e o t h e r hand, an o r b i t of
(x,y) 1
a s required.
is a
C xy = C C xy (z,yFT SES ( z , y F T s
Ts v i a h ( z , y )
sum of a l l elements zy
of t h e form
=
E Cj
and s E S, hsh-l = 8 ,
hE P
permutation group on
0.
T
If
+ + n(CilTr(.C . I .
++
equal t o
Iz E Ci,y
= {(x,y)
T(C.C.)
Since f o r a l l
i < t,
whose k e r n e l i s of t h e r e q u i r e d form.
S, we p u t
t h e union of a l l
1
...,
8).
Thus
1 x 3 E T if and only if
C z
9'3 $7
=
0 for a l l
E {lI...,s}
82
CHAPTER I1
JI
Finally, define
...xF ( S copies)
: FG-'
FX
Czgg
6. =
where
,
C z
i
1Q
< s.
-
Then
a s t h e F-linear mapping given by
(B1 I . . .,BS) J,
T
i s c l e a r l y s u r j e c t i v e and by (21
is
SEqg
w.
t h e k e r n e l of
Hence, by Lemma 3.1,
and t h e r e s u l t follows.
4. INDUCED MODULES.
Throughout t h i s s e c t i o n ,
H
G.
a subgroup of
F
If
V
V,
b u t only a c t i o n of
i s defined on
FH
thus a s an F-module,
vH.
Let FG,
We say t h a t an FG-module
g E G,V.+ G
V
Since w e may consider
... @ Vn
a c t s on t h e s e t
i E {l, ...,n ) .
V
{Vl,.
.. V I
}
V.
with
by t h e r u l e
we s h a l l r e f e r t o t h e set
imprimitivity f o r
V.
V
is imprimitive i f
of F-subspaces
@J
W.
This
can be w r i t t e n a s a d i r e c t
n > 1 such t h a t f o r each
{Vl,
...,Vn}.
'Vi
=
cVl
FG
FH
8.
i s a permutation of t h e s e t
gVi
t o a uniquely
a s a subalgebra of
FH
w e can define an FG-module s t r u c t u r e on t h e tensor product
V = V @
vH
vH*
W be any FH-module.
i s t h e induced module and we denote it by
sum
the
This process w i l l be
c a l l e d r e s t r i c t i o n and it permits us t o go from any FG-module determined FH-module
VH
i s an FG-module, then we s h a l l denote by
FH-module obtained by t h e r e s t r i c t i o n of algebra; equals
a f i n i t e group and
G
denotes an a r b i t r a r y f i e l d ,
gVi
Expressed otherwise,
for a l l
,... lVnl
g E G,
as a system of
The following observation r e l a t e s t h e concept of
i m p r i m i t i v i t y t o t h a t of induction.
4.1. LEMMA. and l e t
ti)
g l , ...,gn
Let
H
be a proper subgroup of
be a l e f t t r a n s v e r s a l f o r
i s a system of i m p r i m i t i v i t y f o r
H
G, in
let
G,
W G and t h e a c t i o n of
F/
with
G
be an FH-module,
g E H.
is transitive.
Then
INDUCED MODULES
(ii) Let
V
be an imprimitive FG-module with
imprimitivity and assume that G the stabilizer of
83
. ,vn}
{Vl,.
{ V l,...,Vn}.
acts transitively on
V1 under the action of G.
v = v 1G.
Then
as a system of
y1
Let H
be
is an FH-module and
n
Proof. (i) By Proposition 1.4, FG
@ g .FH i=1
=
and so
'
i= 1
g E G induces a permutation of the set
Since the left multiplication by { g , 8 W ,... ,gn 8 W } G
(ii) Since
and g ( g 8 W1
g 8 Y, the assertion follows.
acts transitively on
V1, it follows that n in G, say g ,
=
,... ,gn
{Vl,
...,V n }
and H
is the stabilizer of
Thus we can choose a left transversal for H
= (G:H).
such that g.V
=
' 1
Vi, 1
i < n.
Since
and
V as F-spaces, the map
8 : V:+
=
V
g V Q 1
v E V
,
v,
n
However, for each g E G
there exist uniquely determined h E H Thus, for all
... @ g n
given by
n
is a vector space isomorphism.
1
and j E {l,
...,n}
and each
i
E 11,.
such that ggi
=
..,n}, gjh.
we have
as required. Let H let j
:
U+
be a subgroup of G ,
let
V be the inclusion map.
U be a submodule of an FH-module V and Since FG
is a free FH-module
CHAPTER I1
84
(Proposition 1.4), it follows from proposition 1.8.5 that FG module.
is a flat
FH-
Hence the canonical homomorphism
For this reason, from now on we shall identify U
is injective.
.
G
image in V
G
with its
The following lemma collects together some elementary properties
of induced modules. 4.2. LEMMA.
V
Let
V
and
(i) VGc VZG if and only if 1 -
G
(ii) V = 1
(iv)
2
VP
(vl n v ~ = v,") ~n
(vl If V Proof.
V
c
1 -
V G if and only if V
I V ~ + V =~+:v) ~
(iii)
=
V
@
1
V
V.
be submodules of an FH-module
1
V
2
= V
2
VP
then 2'
VG= V G @ Vp
We first observe that (ii) is a consequence of (i), that (v) follows
from (iii) and (iv), and that (iv) is a consequence of the fact that FG free FH-module. To prove (i), observe that Vl VGC VG. 1 -
Let T be a transversal for H
2
5 V2
in G
is a
obviously implies that
containing 1.
Because
vc= E T
@ t 8 V
each element in Therefore, if C ' V 1 -
v2
c t
V G can be uniquely written in the form
V1 E
implies
Vl
18
is such that
V.
E V2.
Thus
.
v1 5 v ~ , proving
To prove (iii), observe that V
V2, then V
Vl E
with vt E
@ Vt
Z%T
G
(i) G 1
+ V G c (V + V 2 ) 2 -
1
.
by virtue of (i)
The
opposite containment being a consequence of the equality
t 8 C u + v l = t @ v +t@V2 1
2
( V E V 1
lPV2
EV,tET)
the result follows. We next provide some important formal properties of induced modules.
4.3. PROPOSITION.
(il
@/A)'
2
Let A
FG*A/FG*B
CB
be left ideals of
FH.
Then
2
85
INDUCED MODULES
(ii) Let I
be a right ideal of FG
such that I s FG*A.
If
V
is an FH-
module, then there is a canonical injective map
Proof.
All tensor products are to be taken over FH.
(i) Consider the diagram
-
o
-FG’-B-FG.B/FG.A
Then the diagram is commutative, and since FG
where all maps are canonical.
a flat right FH-module, both rows are exact, and f , g
are isomorphisms.
is
This
proves the required assertion. (ii) Let
0 is exact.
V-
denote the canonical map
-
FG 63 AV+
FG 63 V
Let x E I , y E FG, and
written in the form q = y a 1
and therefore I ( F G 63 V )
4.4. PROPOSITION.
1
+
V E
The sequence
V.
0
Then xy E I s FG*A, with y
i E FG,ui E A .
which finishes the proof.
Let H be a subgroup of G
sequence of homomorphisms of FH-modules. (il
Then the sequence
a FG 63 (V/AV)&
... + ynan
5.Ker(1 63 U) ,
V/AV.
and let U-
h
V
so q
can be
Thus
. A W be a
Then the following properties hold:
CHAPTER I1
86
0
u
3
-LV J L w-
0
is exact if and only if the corresponding sequence of FG-modules o
-
u
G
~
~
F
~
-
-
-
+
o
is exact (ii) Assume that the sequence
(*)
is exact.
Then
splits if and only if
(*)
(**)
I
splits. (iii) If U
is a submodule of
then $/VG
V,
1
(V/U)G
~
I
Proof. (i) That exactness of (*I fact that FG is a flat FH-module.
implies that of
(**)
is a consequence of the
Conversely, suppose that
(**)
is exact.
Then
Since Ker(1 8 1-I)
=
Im(1 8 A ) ,
it follows from Lemma 4.2(ii) that Kerp
A similar argument shows that Ker(1 8 A ) = 0
Im(l 8 p )
=
8
implies TW
(ii) Suppose that 0y : W-
=
W.
implies KerX = 0
Thus the sequence (*I
A vA
=
TmX.
and that
is also exact.
W-+ 0 is a split exact sequence and let
V be a splitting homomorphism.
18 y
Then
:
8- fi
is obviously
a splitting homomorphism for the induced sequence Conversely, assume that the exact sequence
of FG-modules splits.
Let T
denote a transversal for H in G containing 1.
Consider the mappings
.'cv
V
C t 8 vtl---t ET Then
0
and
T
are well defined FH-linear maps.
splitting homomorphism and let
W
E
W.
Let J,
Then J l C l 8 w ) =
:
V]
84VG
be a
C t 8 vt,vt E
ET
V, and
INDUCED MODULES
c
(1 8 ll)qJ(l8 W ) =
07
t 8 l l ( U t ) = 1 8 2J
ET Thus 2J = ! J ( u )
and therefore
It follows that
T$O
:
v is a splitting homomorphism, as required.
W-
.
(iii) Direct consequence of (i) applied to the natural exact sequence
o~u-v-v/u-o. As
4.5.
an immediate consequence, we derive
COROLLARY.
so is
Let
V
be an FH-module.
fi
If
is completely reducible, then
v.
Another useful property of induced modules is given by If L,H
4.6. PROPOSITION [Transitivity of the induction). with H
5 L,
Proof.
are subgroups of
G
then for any FH-module V,
The required assertion follows from
(+)G
= FG
8 IFL 8 I 4
g
(FG 0 FL) 8
FH
FL
FL
V
FH
gFG8V=VG
FH Let S be a subring of a ring R
r E R,@ E
Hom(R,M), S
define r@
:
and let M
M by
R-
be a left S-module.
(r@)(x) = $[zr).
Given
It is evident
The following that the additive group Hom(R,M) becomes a (left) R-module. S result illustrates how a module of this type can be brought into argument.
4.7. PROPOSITION.
Let
V be an FH-module. Hom(FG,V)
FH
Then
v"
as FGmodules
CHAPTER I1
88
Let [ g
Proof.
,
$ E Hom(FG,VI
define
,,..&n 1
e($)
Be a left transversal for H in G I and for any by
E
=
FH
Then 8
is
.i=1
C gi 8 $(gil) = 0 . Then @ ( g i l )= 0 i=1 Since {gl , . . . , g i l) is a right transversal for H
obviously an F-homomorphism. for all i in G,
n C gi 8 @ ( g i l l .
{1,2,
E
...,n ) .
given g E G ,
and thus 0 elements g
@ E Hom(FG,V)
,...,gn
-1
-1
-1 we have g = hgi
is injective.
-1
Assume that
Because FG
for some h
H.
Hence $ ( g ) = h $ ( g i l ) = O
is a free left FH-module with the
as a basis, for every set
such that @ ( g i l l = U i , l
E
i G n.
{Ui E
V(1
Thus 8
i G n } there is a is an F-isomorphism.
FH Given g E G, there is an hi E H and a permutation i H T ( i ) 11
,...,nl
-1
and hence that g g T ( i ) = 9.h..
such that g . g = hign(i)
as asserted. Let
-1
of Thus
2 2
.
V be an FH-module.
Then the mapping
is an injective FH-homomorphism.
In what follows we shall refer to f
as the
canonical i n j e c t i o n . 4.8.
PROPOSITION (Universal characterization of induced modules).
subgroup of
G, let V be an FH-module, and let f
injection.
Then, for any FG-module
W
and any
:
V-+
8
@ E Hom(V,WH)
Let H
be a
be the canonical there exists one
FH and only one
9 E Homt#,W) FG
which renders commutative the following diagram:
INDUCED MODULES
Since g 8 D = g ( l 8 V ) ,g E G,v E V,
Proof.
is uniquely determined by its restriction to one such $.
v
,...,V n
Let T
form an F-basis of
$ c t 8 vi)
=
J,
fl.
J , (8 t
for some t
{ t 8 Vile E T , 1
fi-
~l
E V.
= t @ ( v ) for any
is an FG-homomorphism.
H.
Since
W
Hence there exists at most
be the F-homomorphism
E T,h E
VG+
in G containing 1 and let
Then the elements
Let $
t @ ( v i ) . Then
write g t = t h
whence
V.
any FG-homomorphism
1 8 V.
be a transversal for H
be an F-basis of
89
W
i
n)
determined by Let g E G
and
Then we must have
$of
= @,
the result follows.
We are now ready to prove the following important result. 4.9.
THEOREM.
Let H be a subgroup of a group G, let V
and let W be an FGmdule. (i) Hom(fl,W)
FG (ii) Hom(W,?)
be an FH-module,
Then
Hom(V,wH)
as F-spaces
FH as F-spaces
Hom(WH,V)
FG
FH Proof. ti) Let f
:
V-
fi
be the canonical injection.
Given
CHAPTER I1
90
Hom(fl,w) FG
Ilr
is an injective F-homomorphism.
-
9
@
If
E
Hom(v,WH)
FH
Pf
Hom(v,WH),
then by Proposition 4.8
FH
there exists a unique 9 E Hom(fi,w) such that
0=
$f.
Thus
8
is an iso-
FG morphism. (ii) Owing to Proposition 4.7, we need only verify that Hom(w, Hom(FG,V) FG FH Given @
E
Hom(W,Hom(FG,V)), FG FH
Then the map @++ @
the map
:
@*
2
as F-spaces
Hom(W ,v) H FH
@* Hom(WH,V) by
define
FH
If I) E Hom(WHH,V), then FH
is an injective F-homomorphism.
W - + Horn(FG,V) defined by
FH
is an FG-homomorphism. Let
Since $*
=
I)I
the assertion follows.
U and V be any FG-modules.
Then the intertmining number for U
and
V is defined to be i(U,V) = dim Hom(U,V)
F Let U
be a completely reducible FG-module, and let V be an irreducible FGWe say that a nonnegative integer m
module.
irreducible constituent of modules contains exactly m Let V/J(V)
FG
U
if a decomposition of submodules isomorphic to
V be an FG-module.
as the head of
4.10. THEOREM.
U
V as an
into irreducible sub-
V.
In what follows we shall refer to the factor module
V.
(Nakayama Reciprocity).
be a subgroup of G.
is the muZtipZicity of
Let F be an arbitrary field and let H
Suppose further that V
W is an irreducible FG-module.
Let
is an irreducible FH-module and
m be the multiplicity of W as an
RELATIVE PROJECTIVE AND INJECTIVE MODULES
p
irreducible constituent in the head of
and let n
as an irreducible constituent in the socle of
WH.
91
V
be the multiplicity of
Then*
In particular, (i) If F (ii) V
is a splitting field for FG
is isomorphic to a submodule of
and FH, then m = n
wH
if and only if W
is isomorphic to
n
a factor module of
v".
We first note that Hom(U,W) = 0 FG U P W. Since for any FG-homomorphism
Proof. with
Hom(fi,W) FG
2
for all irreducible FG-modules U
f
:
... @ Hom(W,W)
. .. @ Hom(V,V)
FH =
V*
WH is
WH, we have
HomIv,V) @ Thus i(V,W 1 H
(rn times)
FG
Since the image of any FH-homomorphism
contained in the socle S of
we have
Hom(fi/J(fi) ,w) FG
z Hom(W,W) @ FG Thus i(fi,W) = m i ( W , W ) .
fi-+W, JCr/;) 5 Kerf
(n times)
FH
n i(V,VI and the result follows by virtue of Theorem 4.9(i).
.
5. RELATIVE PROJECTIVE AND INJECTIVE MODULES. Throughout this section, F denotes an arbitrary field, G a subgroup of
a finite group and H
G.
An FG-module
V
is said to be H-projective if every exact sequence of F G
modules
0-u-
w-
v-0
for which the associated sequence of FH-modules
0
-UH
wH-
vH-
0
92
CHAPTER I1
s p l i t s i s a l s o a s p l i t e x a c t sequence of FG-modules.
The FG-modules
V
is
c a l l e d H-injectiue i f every e x a c t sequence of FG-modules
v-
0-
w-
u-0
f o r which
i s a s p l i t e x a c t sequence of FH-modules, i s a l s o a s p l i t e x a c t sequence of FGThus an FG-module i s p r o j e c t i v e ( r e s p e c t i v e l y , i n j e c t i v e ) i f and only
modules.
i f it i s 1-projective ( r e s p e c t i v e l y , 1 - i n j e c t i v e ) .
5.1. LEMMA.
T
Let
modules, and l e t
H
be a t r a n s v e r s a l f o r
f E Hom(8 ,W 1.
G,
in
Then t h e map
f*
:
W
let
V
V
W defined by
+
and
be FG-
FH
f*(u)
C (tft-llu
=
iET i s an FG-homomorphism which i s independent of t h e choice of
Proof.
Assume t h a t
tut
=
St
t E T with ut E H.
f o r each
C (S@;’)U
Then
C t u t f u t-1 t -1u
=
ET
T.
ET
c tUtUt-1ft-lu
=
E T =
proving that for a l l
f*
f*Wl
i s independent of t h e choice of
T.
I t t h e r e f o r e follows t h a t
g E G. f*(9U)
=
c
c g ( g - l t )f (t-lg)u
c t f t - 3 (gu) =
ET
IET
E g C g - h f ( g - l t ) -$ l%T
as required.
Let
5.2. LEMMA. and l e t
f
:
(VH)l)(i
T be a t r a n s v e r s a l f o r +
V
H
in
be given by
f C C t B u t ) = C tvt %T 7ET
G,
let
V
be a n FG-module,
RELATIVE PROJECTIVE AND INJECTIVE MODULES
Then f is a surjective FG-homomorphism and Kerf
is a direct summand of
It is clear that f is a surjective F-homomorphism.
Proof.
f is in fact an FG-homomorphism, fix g t' E T
93
such that g t = t ' u t .
ut E H
and
G.
E
c
g(
t 8 Ut)
e
=
Then for each
c
g t 8 ut =
c
=
To prove that
E T
there exists
It follows that
ET
ET
t
(6jH.
t ' u t 8 ut
E T
t ' 8 UtUt
ET and so
c
f(g(
t 8Ut))
c
=
ET
t'utut
ET =
=
c
gtut
ET
c
qfC
t
Q Ut),
ET proving that f is a surjective FG-homomorphism. canonical injection.
H n Kerf
Obviously
E(V
= 0.
Let
E :
VH + (V,)
G be the
Because
we conclude that
as asserted.
9
5.3. LEMMA.
Let: T be a transversal for H
and let f
:
V*
in G, let V
be an FG-module,
( V H I G be defined by
c
f(U) =
t 8 t-lu
7ZT
Then f is an injective FG-homomorphism which is independent of the choice of the transversal T and Proof.
Assume
(f(V)),
that T'
is a direct summand of
is another transversal for
each t E 1, there exists ut E H
c t 8 t-lu E T
t'
and =
c ET
E
t'ut
((V,) H
)H.
in G.
T' such that t 8 u,l(t')-$
G
= t'ut.
Then, f o r Hence
CHAPTER I1
94
proving that f
is independent of the choice of the transversa
It follows that for all g E G, V
so that f
is an FGhomomorphism.
Since
WHIG = @ t 8
vfl
*T f
is an injection.
G
(CV,) I H ,
Now
t 8 VH
W' = @
is obviously an FH-submodule of
tfT,#H and we show that
If f t u ) = C t 8 t-lu E f ( V ) n W',
then u = 0.
On the other hand, if
ET
t ' E T n H, we have C t But
ET
-
t 8 t-'t'utl
C
E W'
*T
So the lemma is verified.
The next result establishes several important characterizations of H-projective and H-injective modules. 5.4. THEOREM. migman(1954)). transversal for H
in
Let V
be an FG-module and let T be a
G containing 1.
Then the following statements are
equivalent: (i) V (ii) V (iii) V
is H-projective is isomorphic to a direct summand of is isomorphic to a direct summand of
(iv) There exists
JI
E End(VH)
(V,)
G
8,where
W is an FH-module
such that
FH for all u E V
RELATIVE PROJECTIVE W D INJECTIVE MODULES
(v) V
is H-injective
-
Proof. (i) * (ii): modules 0
U
summand of
.
V'
I
(v,) G.
By Lemma 5.2,
CVH)
-+
submodules splits
where
95
f. V
there exists an exact sequence of FG-
0 such that the associated sequence of FH-
-+
v
By hypothesis,
is H-projective, so
Kerf
is a direct
Thus
V.
(ii) * (iii): Obvious (iv): We first prove that any FG-module V
(iii)
W is an FH-module, satisfies (iv).
It is easily checked that $
Define $
:
of the form V
V+
=
#,
where
V by
is an FH-homomorphism.
Moreover] for all w E FI,
we have
Thus J,
satisfies (iv).
Turning to the general case, we may harmlessly assume
that
#=V@V'
[direct sum of FG-modules)
n
Let
71
:
p-+ V
8. Then
since n
ToJ,
be the projection map and let $ induces an FH-homomorphism V - +
is an FG-homomorphism.
(iv) * (v): Suppose that
U = V 0 W for H
VH
H
some
V, and for all U
E V,
we have
The desired implication follows.
V is an FG-submodule of the FC-module U and that
FH-submodule
W.
and let I$E Hom(U,v) be defined by
F
satisfy (ivl with respect to
Denote by
71
the projection of
UH onto
CHAPTER I1
96
where J1
.
End(VH;i) satisfies (ivl
FH U
=
V @ Kerf$.
But, by Lemma 5.1,
FG-submodule.
If v E V,
f$
(v) * (i): Assume that
V
is an FG-homomorphism, so KerG
Then, by Lemma 5.3,
is H-injective.
isomorphic to a direct summand of
(VHl
infer that there exists $ E End(VH) FH
G
.
v+
v
and define
0
:
U
4
u
=
G/
(direct sum of FH-modules) Set
4
= f-l
U by
6
B(U) @Kerf
5 . 5 . COROLLARY.
= d
flW is an FH-isomorphism of W onto VH.
proving that U = e ( U )
Thus
is
is a surjective FG-homomorphism for which
6(u)
Then, by Lemma 5.1,
V
such that
U=Kerf @ The restriction f =
is an
Invoking implication (iii) * (iv), we
(1 t$t-l)v E T :
and thus
V is H-injective.
Hence
Assume that f
then $(U) = u
I: t0Jlft-lu ET
is an FG-homomorphism and
+
Kerf.
Finally, if
B C U ) E 8 ( U ) n Kerf,
then
and the result follows.
(Higman, 1954).
divide the index of H
=
in G.
Assume that the characteristic of
F does not
Then any FG-module is H-projective and H
-
97
RELRTIV’E PROSECTIVE AND INJECTIVE MODULES
injective. proof.
Then J,
V
Let
be an FG-module and let $
4 V
V
:
be defined by
is an FH-homomorphism and for all U E V
c c t$t-l)u
= u
*T where
is a transversal for H
T
Let N
5.6. COROLLARY.
fi
and let V
G
be a normal subgroup of
be an irreducible
N
in G ,
is completely reducible.
VG and let T be a transversal for N
Let W be a submodule of
Proof. in G.
Now apply Theorem 5 . 4 .
If the characteristic of F does not divide the index of
FN-module. then
in G.
Since
Cs;,,= (#)N
it follows that
reducible, and thus WN
Hence WN
is completely reducible.
(#IN.
is a direct summand of
Consequenctly, W
N-injective.
@ t @ V ET is completely
By Corollary 5.5,
is a direct summand of
r/;
W is
and the result
follows. Let
U
be FG-modules with F-bases u
Then U 8 V
respectively.
{ui 8 u .I 1
V
and
1
,...,u
is a vector space over F
and U , . . . , U r n , 1
with an F-basis
F Q
i
G n, 1 & j G m}.
We can define an action of
G
on
g(ui8 u .I 3
for all i , j
=
g u . 8 gv
and g E G, and then extend to
z
j
U 8 V and FG by linearity.
F FG-module U 8 V
is called the (inner) tensor product of
F 5.1. PROPOSITION.
Let
s
U 8 V by
F
3
5H
be subgroups of
U
and
V
The
.
G and let V,W be FG-modules.
WH is 5’-projective, then V 8 W is S-projective F (iil If W is projective, then V 8 W is also projective. F Proof. (il Let X be a transverse1 for H in G and let Y be a (i) If
V
is H-projective and
CHAPTER I1
98
s
transversal for
in H.
Then XY
in G.
is a transversal for S
By
Theorem 5.4, there exist y E End(vH)
and
6
and
C (y6y-l)w L8y
Endtkls) FS
FH such that
(
c
%?yola:
-1 )v =
2,
3SX for all
Then y 8 6 E EndC(V 8 W ) s ) FS F
W.
E v,W
2,
=
w
and a straight-forward
calculation shows that
(
c
q c y 8 6)y-lx-l)
It,8 W )
= t,
8 w
sx,yEy W.
for all U E V,w
The desired conclusion is now a consequence of Theorem
5.4. (ii) Apply ti) for the case H = G
and
S = 1.
9
The following lemma collects together some simple properties of H-projective modules. 5.8. LEMMA.
Let
S 5 H be subgroups of G and let V be an FG-module.
n
8 Vi, where Vi is an FG-module, 1 i=1 projective if and only if each V. is H-projective cil Let V
(iil
If
=
is S-projective, then V
V V
(iii) If
is H-projective and
(ivl For any FH-module W,
i
n.
Then
V is H-
is also H-projective
VH is S-projective, then V is $-projective
the FG-module
8
is H-projective
(vl If kr
is an S-projective FH-module, then any FG-module which is a direct
summand of
d'
"
Proof.
is S-projective.
(i) Assume that
V is &projective.
which satisfies Theorem 5.4Iiv). onto Vi
defined by the decomposition V =
implication (ii) onto Vi,
Let nil
* (iiil of Theorem
5.4,
1
@Vi.
Then there exists $ E End(VH)
FH 6 n, be the projection of
V
Then, as in the proof of
n.03, induces an FH-homomorphism of
which also satisfies Theorem 8.4(.iii).
Thus each
vt Vi is H-projective.
Conversely, if each Jli E EndlViIH satisfies Theorem 5.4.Uv1, then $ E End(VH1
FH
FH
RELATIVE PROJECTIVE AND INJECTIVE MODULES
99
$ l V . = $i, obviously satisfies the same condition.
defined by
-
V
Hence
is
H-projective. (ii) Let 0
U
W-
--L
V -
* UH
the associated sequence 0 sequence 0
us
-+
sequence 0
U
+.
ws
-+
-+
1
VH.
Thus
< be a cyclic group of order p
n
.
Owing to Corollary
2.4, we need only verify that t(G) =
By Proposition 1.1, J ( F G ) = IIG)
i elements g -1, 1
i c p".
=
FG(g-11.
and so J ( P G )
is the F-linear span of all
Since
gi-1 JPGI
pn
=
(g-1) u+g+. * . +gi-1 )
The desired conclusion is therefore a consequence of the
equality
applied for k G n .
.
k g p -1
k =
Ig-lP
3 . A CHARACTERIZATION OF ELEMENTS OF
J ( F G ) : THE GENERAL CASE
Let G be a finite group and let F
be an arbitrary field of characteristic
p > 0.
The problem that motivates this section is to discover necessary and
sufficient conditions for an element z
=
zz
certain additional hypotheses le.g. when G
99 E FG
to be in J ( F G ) .
Under
is p-solvable), these conditions will
be simplified in Sec. 5. To accomplish our aim, we shall have to set up some machinery pertaining to the theory of Frobenius and symmetric algebras.
Many of the results presented
below will only be needed in subsequent investigations.
We present them here in
order not to interrupt future discussions at an awkward stage.
Our method
n CHRRACTERIZATION OF ELEMENTS OF J ( F G ~
123
borrows heavily from an important paper due to ~ulshammer(1981a). Throughout this section, unless explicitely stated otherwise, A finite-dimensional algebra over an arbitrary field F
denotes a
and A* = H o m ( A , F ) .
The
F left (right) A-module A Given a subset X
means the regular left (right) A-module.
of A ,
define A X
and XA
A X = {Caizilai E A , Z ~E
by
XI
XA = {Cr.a.Iai E A,zi E XI 2 2
If
X = {a)
tively. all a E A
(A,$)
that the pair pair
(iii) a
(ii) J,(Aa) = 0
0
=
We shall refer to A
A
and XA,
respec-
the following conditions are equivalent:
$(ad)
(1)
instead of A X
is said to be a Frobenius d g e b r a if for
where $ E A * ,
(A,$),
pair
A
and aA
then we shall write A a
(A,$),
=
0
itself as a Frobenius algebra if there exists J, E A*
such
is a Frobenius algebra. where
is said to be a symetric aZgebra if
$ E A*,
(A,$)
is a Frobenius algebra such that for all r , y E A
J,(zy) = $ ( p )
The algebra A (A,$)
is called a symmetric dgebra if t ?re exists
E A*
such that
is a symmetric algebra.
Let f
: A
x A+
F
be a bilinear form.
Then f
is called nonsingukm if
the following conditions are equivalent. (i) f ( a , A )
=
(ii) f ( A , a ) = 0
0
We shall refer to f
(iii) a = 0
as being associative if for all z,y,Z E A
and symmetric if
f(2,Y)
3.1. LEMMA.
(i) Assume that
(A,$)
=
f(y,z)
i s a Frobenius algebra.
for all z,y E A
Then the map
CHAPTER I11
124
f
:
A x
form.
defined by Q(x,y) = $(.y) is a nonsingular associative F-bilinear
F
A -+
Furthermore, if
(ii) Assume that f : A x and let $ : A+
F
is a symmetric algebra, then f
(A,$)
A-
F is a nonsingular associative F-bilinear form $(x)
be defined by
Furthermore, if f
algebra.
= f(2,l).
is symmetric then
(i) For any x , y , z E A ,
Proof.
=
AfCcc,y).
Suppose that f @ , y ) = 0 I=
0.
A
Then (A,$)
is associative.
is a symmetric algebra.
Given h E F,
A.
we also have
Then $(d) = 0
similar argument shows that if f b , y ) = 0
proving that f
is a Frobenius
Hence f is an F-bilinear form on A.
for all y E
proving that f is nonsingular.
(A,$)
we have
and similarly f(rc,y+z) = f h , y ) + f & , z ) .
and similarly f ( z , h y )
is also symmetric.
and therefore
for all x E A,
then y = O ,
For all x,y,Z E A, we have
Because $key) = $(yr)
implies f ( z , y ) = f ( y , x ) ,
the assertion follows. Lii) It is clear that $ E A * .
f(ab,l)= 0 of f a = 0,
for all b E
A.
implies that a = 0 . proving that
@,$I
Let a E A
Then
However, f(ab,l) = f ( a , b ) , so the nonsingularity A similar argument shows that if $(/la) = 0 is a Frobenius algebra.
for all x,y E A
as asserted. Note that A*
be such that $(d) = 0.
is an (A,Al-bimodule via
If
then
f is symmetric, then
A CHARACTERIZ?iTION OF ELEMENTS OF
for a l l
x,a E A
3 . 2 . LEMMA.
f E A*.
and
(i) Assume t h a t
i s a Frobenius algebra.
(A,$)
f defined by modules.
f ( a ) (3)
125
J(&'G)
$(xu) f o r a l l
=
Furthermore, i f
:
Then t h e map
A-A* i s an isomorphism of l e f t A-
Z,a E A ,
is a symmetric a l g e b r a , then
(A,$)
f
i s an iso-
morphism of ( A , A ) -bimodules.
(iil
f
Assume t h a t
$(a1 = f t l ) ( a ) .
:
A+ (A,$)
Then
i s an isomorphism of l e f t A-modules and s e t
A*
is a Frobenius algebra.
(A,$)
isomorphism of (A,A)-bimodules, then
(i) I t is obvious t h a t f o r a l l
Proof.
for a l l
x,y E A .
proving t h a t
f
f
If
r,x E A ,
a E A , f ( a ) E A*
dh4 = did*
,f
By t h e d e f i n i t i o n of
$,
is a b i j e c t i o n .
F
i s a symmetric a l g e b r a .
(A,$)
and t h a t
then
F Assume that
f i s an
i s a symmetric a l g e b r a .
i s a homomorphism of l e f t A-modules.
i s i n j e c t i v e and s i n c e
Furthermore, i f
Then, f o r a l l
r,x E A,
w e have
a s desired.
$ E A*.
(ii) I t is c l e a r t h a t
f o r all
x E
f o r all
x
X and hence a
E A,
Assume t h a t
w e have
then
f
a
= 0.
=
Suppose t h a t
0.
$(ax)
= 0
for a l l
z E A.
A similar argument proves t h a t i f
Hence
(A,$)
Then
$(xu) = 0
i s a Frobenius algebra.
is an isomorphim of (A,A)-bimodules.
Then, f o r a l l
x,y E A ,
126
CHRPTER 1 x 1
.
as required.
3 . 3 . COROLLARY.
The following conditions are equivalent:
A is a Frobenius (respectively, symmetric) algebra
(i)
(ii) There exists a nonsingular associative F-bilinear form f :
A
x
A+
F
(respectively, there exists a nonsingular, associative, symmetric F-bilinear form
f
:
A x A+
F)
(iifl A
A*
A
as left modules (respectively,
Y
A*
.
as (A,A)-bimodules)
Direct consequence of Lemmas 3.1 and 3.2.
Proof. 3.4. LEMMA.
Let
A ,Az,...,An be F-algebras.
n nAi
Then
is a Frobenius
i=1
(respectively, symmetric) algebra if and only if each
Ai
is a Frobenius
(respectively, symmetric1 algebra. Suppose that $ . f5 A? is such that ( A i , q i ) w z z and define $ : AF by put 4 =
Proof.
is a Frobenius algebra
nAi
i 6 i G n.
i=1
Then
is obviously a Frobenius algebra.
(A,$)
Frobenius algebra, then so is of $
onto Ai.
(Ai,$il,
1
Conversely, if
i G n, with qi
Finally, for a = Ca ,...,an) and b = (b
if and only if $i(uib2) = $icbz&i)
for all
i E 11,.
..,n}.
(A,$)
is a
being the projection
,...,bn),$(ab)=$(ba) Thus the lemma is
proved, We have now accumulated all the information necessary to prove the following important result. 3.5. THEOREM. (Eilenberg and Nakayma (1955)).
If
(i1
(iil
If
A
1
and
A
'F
A is a semisimple F-algebra, then A is symmetric
Proof.
(i) Suppose that
A
1
and
A
are Frobenius F-algebras.
corollary 3 . 3 , there exists an isomorphism fi :
i
= 1,2.
@A
are Frobenius (or symmetric) F-algebras, then so is A=A
Consider the map
f
:
Ai-
A d A* defined by
Then, by
Atz of left Ai-modules,
f
A routine calculation shows that
over, if each
is an isomorphism of left A-modules.
fi is an (Ai,Ail-isomorphism, then f
More-
is an (A,A)-isomorphism.
The desired assertion is now a consequence of Corollary 3.3. (ii) We first assume that A
=
trace map, we conclude that A
Mn (F) for some n 2 1. If D
is symmetric.
Then, by looking at the is a division algebra over
F, then
Invoking Ctl and Lemma 1.4, we may therefore assume that A
J,
Then, for any nonzero
U,$I
in A * , ( A , $ )
is a Frobenius algebra.
J,
in A*.
To prove that
J,(IA,AI) = 0
is a symmetric algebra, it suffices to verify that
nonzero
is a division algebra.
The latter will follow provided we show that
for some
[A,Al # A.
The last statement being independent of the qroundfield, we may assume that F the centre o f Let A
E
E
we have
A. F which is a splitting field for AE.
be a field extension of
Mn ( E l for
is
n 2 1.
some
IAE,AE]
# AE.
For a given subset
Since each matrix in
[AE,AEI = [ A , A I E ,
But
X of A ,
we denote by
so
Then
[Mn&) ,MnlE)l has trace 0,
IA,AI #
A,
as required.
RCY) and r(X) the left and
right annihilators of X defined by
rCYl
It is obvious that
E(X)
ly.
X is a left (respectively, right) ideal of A,
Moreover, if
and
are left and right ideals of
(A,$)
Define the subsets
'X
is a Frobenius algebra and that X and 'X
'X
X It is plain that if
1
of
A
E
then
A($(Xd
X is a subspace of A,
&(XI
is a subset of
by
= {a E A I J 1 W ) = =
respective-
A.
(respectively, r ( X ) ) is a two-sided ideal of Suppose that
A,
=
0) 01
then so are
'X
and
X
L
.
A.
CHRPTER I11
128
3 . 6 . THEOREM.
of
(A,$)
Let
be a Frobenius a l g e b r a and l e t
X
be an F-subspace
A.
(i) (IX)'
(X',
c
=
X
1 dimXl = dim X = d i d - d i d
and
F
X
(ii) If
F
F
i s a l e f t ( r e s p e c t i v e l y , r i g h t ) i d e a l of
R(X) =
(respectively,
I
,
= (1-e)A
(Ae)'
F
x).
A,
then
r(X) = X
I n p a r t i c u l a r , f o r every idempotent
A/(Ae)'
3
eA
X
dimX F If
X
(iv)
A,
i s a l e f t i d e a l of
+
F
X
The mapping
then
dlmr(X) = dim.4
F
i s a r i g h t i d e a l of
A,
A.
of
a s r i g h t A-modules and dimeA = d i d e .
F (iii) I f
e
1
&@(XI) = X
and
F then
k rCY) i s a d u a l i t y of t h e l a t t i c e of l e f t i d e a l s of
A
onto t h e l a t t i c e of r i g h t i d e a l s , t h a t i s
X1,Xp
for a l l l e f t ideals Similarly,
X
n R(X)
of
is a d u a l i t y of t h e l a t t i c e of r i g h t i d e a l s of
t h e l a t t i c e of l e f t i d e a l s of (v)
If
(A,$)
If
(A,$)
Cartan matrix of Proof.
X
of
onto
A.
A.
F
i s symmetric and
A
X.
is a s p l i t t i n g f i e l d for
A,
then t h e
i s symmetric.
(i) Fix an F-basis
is an F-basis of
A
is symmetric, then
f o r any two-sided i d e a l (vil
A.
The map
...,an 1
{al ,
of
A
such t h a t
~U1,...,U
m
},m
n
J(FG)
A CHARACTERIZATION OF ELEMENTS OF
XI consists
i s an F-isomorphism such t h a t t h e image of
A
1
=
1 = 2
... = 1rn = 0.
1
Hence
d i d F I t follows t h a t
I
dim X = d i d - d i d . F F F
did-dimX F F
=
129
of a l l
(Al,.
../In)
with
and, by a similar argument,
dim?'X)l = d i d F F
1 1 t X) 3 X, we i n f e r t h a t
Because
tX)'
=
X,
as required.
The second e q u a l i t y
i s proved s i m i l a r l y .
X is
(ii) Suppose t h a t
$(ax1 = 0 or i f and o n l y if $CaXAl = 0. 0,
R(X1 = 'X.
to
aX
=
of
A
follows by the same argument.
we have
x
E (&)'
Then
A.
a r i g h t i d e a l of
i f and only i f
Since the l a t t e r e q u a l i t y i s equivalent
The e q u a l i t y
= 0
P(X) =
for all
NOW
* Aex
a E 'X
* ex
= 0
1
X
for a l e f t ideal
X
x e A.
*x
E (l-e)A,
so
Ue)' = Il-e)A dfmeA = d i d e ,
Finally
by applying (i)
F
F
and
bii) Direct consequence of (i) and
A/&?)'
eA
.
ui).
It is clear t h a t
(iv)
Invoking C i i i ) , w e conclude t h a t
The correspondkng a s s e r t i o n f o r l e f t i d e a l s follows s i m i l a r l y .
(v)
u,$) i s a
A s s u m e that
X of
A.
,...,en
Let
e
let
C = Cc..)
1
Then
'X
=
XI f o r any s u b s e t
Now apply (ii).
Assume t h a t
(vil
symmetric algebra.
ZJ
(A,$)
is symmetric and that F i s a s p l i t t i n g f i e l d f o r
be a complete s e t of orthogonal p r i m i t i v e idempotents of be t h e Cartan matrix of
A.
Then
A
A. and
CHRPTER 111
130
n
n @ezJl
A = @Ae.= i=1 z
and therefore
n
A = Fix j,k E 11
,...,n}
with j # k .
@
(direct sum of F-spaces)
eAe
i,j=l 3 ; Then
and so, applying (i)and Proposition 1.10.16(1),
c.. = dime .A@ = d i d 3% p 2 j F
we have
- dim(e A e .) I F
z
3
A similar argument shows that c . . G c . . and the result follows. $3 3% We next concentrate on factor algebras and prove the following result. Let X be an ideal of A.
3.7. THEOREM (Nakayama(1939) I .
(il
Assume that
(A,$)
i s a Frobenius algebra.
algebra if and only if there exists a E (iil
Assume that
(A,$)
Then A/X
A such that rlX)
i s a synnnetric algebra.
is a Frobenius =
aA = Aa.
Then A/X
i s a synnnetric
algebra i f and only if there exists a E z(A) such that P(X) = ud. Proof. II : A
3
Ci)
A/X
Suppose that
(A/X,l.l) i s a Frobenius algebra, and let
be the natural map.
exists a E A
such that PIT = $,
show that r(X) = aA = Aa.
Then, by Lemma 3.2Ci) and Corollary 3.3, there where $a(b) = $(ba)
b
E
RCAa).
Thus
b
E
.
Since Jla(X) = $(Xu) = 0 ,
fore ad C_ X1 and, by Theorem 3.6Cii),
aA
5 r(X).
for all b E A . 1 we have a E X
Suppose that
b
We shall There-
€ &(&)
or
Then ba = 0 and hence
X and so both C(u.4)
we infer that
aA
=
and
~ ( A u ) are in X.
r(R(aA)l 2 rU1, proving that rlX)
Applying Theorem 3.6(iii)r =
ad.
Consequently,
A CHARACPERIZATION OF ELEMENTS OF J ( F G )
Aa
5 r(X)
and therefore LUa)
3.X.
It follows that
X
=
131
RUd
and thus
r ( X ) = rR(Aa) = Aa. Conversely, assume that r ( X ) = a4 = A a
Because a E r ( X ) , p
But r(X) =
b E k(u.4).
Suppose that
Then $(Aha) = 0, so ba = 0 and thus
aA implies that X
hence b E X.
= k ( r ( X )) = k ( d ) ;
A similar argument shows that p (Cb+X)( A / X ) 1 = 0
A/X
Define
is well defined and obviously an F-linear map.
is such that V((A/X) (b+X)) = 0.
bE A
for some a E A .
implies b E X, proving that
is a Frobenius algebra.
(ii) Suppose that A / X
By ti), r C X ) = aA
is symmetric.
for some a E A .
On
the other hand,
for all r , y E A .
for all y E A ,
Hence ay = y a
.
Conversely, suppose that a E Z U ) metric.
and so
Then, by Cil
,
a E Z(A).
A/X
is obviously sym-
So the theorem is proved.
We next exhibit some important properties of socles and commutator subspaces of Frobenius and symmetric algebras.
First, however, it will be convenient to
recall the following piece of information, Then the socle of
V,
defined as the sum of all irreducible submodules of
V.
Let
V
be a left A-module.
module is defined similarly. socles of AA
and A A ,
3 . 8 . THEOREM.
SocA
Assume that A
are defined as the
In case the left and right socles of A
to denote both of them. i s a Frobenius algebra.
ti) The left and right socles of A
SocV, is
The socle of a right A-
The left and right socles of A
respectively.
coincides, we shall write
written
coincide.
Furthermore,
132
CHAPTER 111
dim s o d = dim(A/JCA)) F F (ii) For each primitive idempotent e E A,(socA)e suhnodule of Ae. (iii) If A
In particular, soc(Ae) = (SocA)e
is symmetric, then such that SocA = Az
(a) There exists z E Z U )
(c) For any irreducible A-module
socle of A.
Let e
1
Ae/J(A)e
V, V is isomorphic to a submodule of P ( W .
Proof. (i) and (ii): Setting J = J ( A ) , is the left socle of
and J(A) = L(Z)
A, soc(le)
(b) For any primitive idempotent e E
that r ( J 1
is the unique irreducible
A.
it follows from Proposition 1.6.29
By a similar argument, !.(J) Then Je
be a primitive idempotent of A .
By Theorem 3 . 6 , the map X C +
A containing
and the set of right tdeals of
inclusion-reversing bijection
X W X'/(Ae)l
and the set of submodules of A/(Ae) inclusion-reversing bijection
eA.
which implies that e(Je)'
.
XI-+ ex'
and the set of submodules of eA. ducible submodule of
.t
er(J).
is an inclu-
1 CAe)
.
we thus obtain an
between the set of submodules of Ae
Again, applying Theorem 3 . 6 , we obtain an between the set of submodules X of Ae
In particular, e(Je)'
Further, for y E A ,
=
'X
X of A contained in
sion-reversing bijection between the set of left ideals Ae,
is the unique
i s an irreducible module
maximal submodule of Ae, and the quotient Ae/Je (Propositions 1.9.4 and 1 . 6 . 3 6 1 .
is the right
Thus er(JI
is the unique irre-
we have
is the unique irreducible sub-
module of eA. Because J rCJ)J
annihilates all irreducible A-modules, we have e r ( J ) J = 0 .
is annihilated by all primitive idempotents e E
that r ( J ) J = 0 , symmetry.
and therefore r ( J )
5R(J).
A.
Thus
The latter implies
The reverse inclusion holds by
The furthermore assertion being a consequence of Theorem 3 . 6 ( i ) ,
property ti) is established.
The same argument (reversing left and right) also
proves (ii).
(iii) Note that Ca) is a consequence of Theorems 3.5(ii)
, 3.6(iii)
and 3.7111).
A CHARACTERIZATION OF ELEMENTS OF J ( F G )
To prove (b), let $ E A*
be such that
133
$s a syrmnetric algebra.
[A,$)
foregoing, r ( J ) e is the unique irreducible submodule of Ae
By the
and J r ( J ) e = 0.
Hence every f E HomCAe,r(J)e) induces a map A E Hom(Ae/Je,r ( J ) e ) . Since A A Hom(Ae/Je , r ( J ) e ) 2 e r ( J ) e as additive groups (Lemma 1.5.5), we are left to A
verify that e r ( J ) e # 0 . Assume by way of contradiction that e r ( J ) e = 0.
and hence r ( J ) e = 0 ,
which is impossible.
To prove (c), observe that V idempotent e
This proves (b). and P ( V )
Ae/JCA)e
of A(propos$tion 1.9.4)
.
Then
By (b), V
Ae Socue)
for some primitive
5 Ae
and the
result follows. As a preliminary to the next result, we prove 3.9.
LEMMA.
(i) Assume that
[A,$)
is a Frobenius algebra.
is the largest left ideal of A
set X of A, E ( X )
contained in ' X
is a symmetric algebra. Then Z ( A ) 1 1 Proof. (i) By the definition of X, we have $(XI 5 x .
(ill Assume that
left ideal of A
so
IX
=
(A,$)
contained in
0 and thus
I5
'x.
Then, for any sub-
=
IA,Al
1
Let 1 be any
Then
?,(A'), as required.
(ii) For any z E A, z E [A,A]'
if and only if for all z,y E A
or if and only if
Since the latter is equivalent to yz-zy = 0 for all y E A, established.
Fran now on, we assume that c h F
=
p
3
0
and s e t
the result is
CHARACTER I11
134
TU) is
Then
A
an F-subspace of
containing
we a r e now ready t o prove
[A,AI.
t h e following important r e s u l t .
3.10. THEOREM (Kbhammer (1981all
. With
the n o t a t i o n above, the following
p r o p e r t i e s hold: (i) T ( A ) = [A,AJ
+ SU)
and
JU) is
t h e l a r g e s t l e f t i d e a l of
A
contained i n
TU).
(iil
A
If
A
(iii) If
T(A)
(a)
1
(bl S o d
= ZU)
('2 (A)
=
A
is a symmetric a l g e b r a , then
I
(TU)1'
(c)
1
S o d = T(A)
is a Frobenius algebra, then
n SocA and, i n p a r t i c u l a r ,
n
TCA)'
i s a n i d e a l of
ZU)
socA)A
= ZcB1l @
.. . d Z ( B P ) ,
B 1 ,. ..,By a r e a l l blocks of
where
A
which a r e sfmple F-algebras. (d)
F is a s p l i t t i n g f i e l d f o r A ,
If
a r e simple a l g e b r a s is equal t o
then t h e number of b l a c k s of
A
that
dim(T(Al')2
P (i1
Proof. any
ZU)
z E
both
[A,A]
W e f i r s t treat t h e case where
and
and
E
is a simple algebra.
T(A)
can be regarded a s a l g e b r a s over t h e f i e l d
Z U 1 = F,
i.e. that
be t h e a l g e b r a i c c l o s u r e of
F
A
Mn(E)
F
n P
1.
Then c l e a r l y
TUE)= IAE,AEl
and
1
Because
dim[A,Al = d%IAE,AEl F
Z(A).
We may
is a c e n t r a l simple F-algebra.
so t h a t
AE = E @ A f o r some
Since f o r
x,y E A,
t h e r e f o r e assume t h a t Let
A
and d i m A = dim AE
F
E
135
,
then since dim[A,A] = d i m A - 1. If it were t r u e t h a t A = T ( A ) F F i s an E-subspace of AE w e would have AE = T [ A E ) , a c o n t r a d i c t i o n .
we i n f e r t h a t
T(A )
E
Hence A # T ( A 1 (A,$)
[A,A]
and t h e r e f o r e , by t h e above,
Ker$
and hence
proves the case where
T(A)
A
c o n t a i n s no nonzero l e f t i d e a l s of
H = A
1
Ai
< i < m,
A' 1
=
i s simple,
But then
A.
This
and write
A/J(A)
@...@Am
are pairwise orthogonal simple F-algebras.
Then,
T ( Az. )
=
[Ai,Ail,
by t h e s p e c i a l c a s e proved above, and t h u s
~ b v i o u s l y IA,AJ
+ J U ~c- T ( A )
and
T U ) / J ( A )5
T G l = T(A)/J(A) and so TlAl = IA,Al Assume t h a t
T
is a l e f t i d e a l of
contained i n
+ JU).
.
A
contained i n
T(A)
Tfl)
and hence i n
[z,x].
i s a l e f t i d e a l of
A'
~ 2 1 . Consequently,
is a symmetric a l g e b r a (Theorem 3 . 5 ( 1 1 1 ) ) .
semisimple, argument,
A
is a simple algebra.
Turning t o the g e n e r a l case, s e t
where t h e
Since
IA,Al.
=
J, E A* (Theorem 3.5(11)1.
is a symmetric a l g e b r a f o r some
C,
!?(A)
Iz,z]
c o n t a i n s no nonzero l e f t i d e a l s of
A'.
Then ( I + J ( A )1 / J ( A ) Since
is
Thus, by t h e previous Hence
I
5 JL 4 )
and
the r e q u i r e d a s s e r t i o n follows.
(ii) A s s u m e t h a t A
By ( i ) and (l),
JUl
Applying Lemma 3 . 9 ( i l ,
is a Frobenius algebra.
Thanks t o Theorem 3.6, w e have
1 i s t h e l a r g e s t l e f t i d e a l of
A
contained i n
(T(A)l)
we conclude t h a t
1
J ( A ) = L(T(A) Invoking (2) and ( 3 ) , it follows t h a t
1
= t(T(A) A)
(3)
C W T E R XI1
136
SocA =
1
r(J(A)I = T ( A ) A ,
a s required.
A
(iii) A s s u m e t h a t
i s a s p e t r i c algebra.
By (ii)and Lemma 3 . 9 ( i i ) / t o
prove (a) w e need only v e r i f y t h a t
Because
1 R(T(A) A )
A,
is a l e f t i d e a l of
we have
Consequently,
Property (b) i s a consequence of ( a ) and (ii), while ( d ) follows
proving ( a ) . from ( c ) .
W e a r e therefore l e f t t o v e r i f y ( c ) .
To prove ( c ) , w e may harmlessly assume t h a t
A
i s indecomposable, i n which
case it s u f f i c e s t o v e r i f y t h a t
J U ) = 0.
Suppose f i r s t t h a t i d e a l of
J ( A ) # 0.
ZU)
, so
Then
TU)'
19 S o d
Then
= ZL4)
and so
1
1 E SocA n Z ( A ) = T ( A )
1 T(A)
5 SocA
1
1 E ?"(A)
is a l o c a l algebra, we have
However, by ( a )
1 IT(A) 3
and thus
and thus
.
*
= Z(A)
.
.
But
TU)'
i s an
Next assume t h a t
Taking i n t o account t h a t
Z(A)
A CHARACTERIZATION OF ELEMENTS OF J ( F G )
as required.
137
.
We have completed the development of the general machinery, and we are confronted with the problem of characterizing the elements of J W G ) , is a field of characteristic p
where F
0.
Define a map
tr
:
FG
F
-+
called the truce, by
The following observation will enable us to take full advantage of the results so far obtained.
CFG,trl
3.11. LEMMA.
Cil
(ii) Let e
be an idemptent of FG
eFGe.
CeFGe,tr'l
Then
Proof.
y
=
Z y#
is a symmetric algebra and let tr'
be the restriction of
tr
to
is a symmetric algebra.
(i) It is clear that tr E HomCFG,F). F i n FG. Then we have
Fix x =
C x# SEG
and
SEG
trm) =
C x
=
$G
C y x -1 = tr Cyx) @Gg g
and for all g E G In particular, if t r ( F G x ) = 0 then x = 0 as required. Uil
Assume that y E e F G e
is such that tr' ( e F G e y ) = 0.
and
prwing that y = 0. For any subset
S o the lemma is true.
X of G , we write
x-1 =
{ s - l l x E XI,
x+ = c XE
x
x
Then y
=
eye = y e
CHAPTER 111
138
and denote by FX the F-linear span of X. Let C = 111, C 2 , 1
section of
G
...,Cr
be all p-regular classes of G .
By a p-regutar
associated Mith Ci, we understand the set
It is an easy consequence of the definition that
r with Si n S. = @ for i # j, each Si is a union of conjugacy (i) G = u S i=1 i 3 is a union of conjugacy classes of p-elements of G. classes of G and S (ii) An element C E Ck(G1
i s a p-regular section if and only if
C
is of p-
defect zero. Let X be a subset of G
3.12. LEMMA.
(i) tr(ux*) =
C u g E FG.
and let u =
SfG
c
and, in particular, tr CuSil =
Proof. By the definition of the product of elements in F G ,
Hence, by the definition of the trace map, tP(uX')
uX+
=
=
E
u -1 =
SEX $ 7
we have
C
u
6 x - 1 9'
Also
0 if and only i f for all t E G Cu
&X
=
tg-1
c
= o
u
FX-1 tg
as required. The following result will enable us to take full advantage of Theorem 3.10(iiil 3.13. LEMMA.
...,Sr
Let S1,S2,
(i) T V G I = { C
x g E FGI C x 9-i
be all p-regular sections of G . =
Then
0 for all i E {l , . . . , r } }
gEG l r (ii) T V G ) = C FS; i=1
Proof.
I*)
ti) We first observe that there exists n 2 1 such that
m gp
for all m
=
9P' Indeed, write / G I = pak with n E N such that pn
E
lhcd k).
Cp,k)
O(mod n ) = 1.
and all g E G
Because
(p,k) = 1, there exists
Replacing n by its multiple, if necessary,
CHARACTERIZATION OF ELEMENTS OF $@‘GI
A
we may assume that n 3 a.
Suppose that m
rn
Then
$
139
is divisible by
n and let g E G.
m
= 1
6, gp,.
and, since p m z lbcd k),
=
Thus
proving ( * ) .
m
x
x# E FG. Then, by ( * I , we may choose m 2 1 such that 9EG m for all g E G and such that x E TVG) if and only if $ E [FG,FG]. Let
=
8
=gpl
Wing to Lemma 2.2.1,
m
and thus x E T V G )
c
if and only if
2.2.2, that x E TWG)
$’“g E [FG,FGI. It follows, from Lemma SEG P’m if and only if C 2 = 0 for all i E (1 ,...,I-}. SEGi g
Since
the required assertion follows.
r (ii) By Lemma 3.12(i)
and (i),
c
FS;
5
1
i= 1 x = 9EG
E T(FG)’
regular section 5‘;’
and let a , b E Si.
.
Conversely, suppose that belong to the p -
Then a-1 and b-l
and hence, by (i), a-l-b-’
E !Z‘VG).
But then
tr(za-l-xb-l) = 0 which ensures that
proving (ii). We are now ready to prove the following important result. 3.14. THEOREM(Brauer(1955), Tsushima(1978al). Let F istic p > 0 and let S ,S2,...,Sr
Proof.
be a field of character-
be all p-regular sections of
By Theorem 3.1Otiii) (a),cb) ,
G.
Then
CHAPTER 1x1
140
r Invoking Lemma 3.13(ii), w e conclude that SocFG =
+
C PG*Si. i=1
It follows from
Proposition 1.6.30 and Theorem 3.8(iii) that
J(FGI
= =
r I ~ C S ~ ~ =I?( F G )c FG-S~) r i=l n I~(FG-s~) i=1
The desired conclusion is now a consequence of the equality k(FG*S;)
+
= R(Si).
’
We have now come to the demonstration for which the present section has been developed. 3.15. COROLLARY.
p
7
Let G be a finite group, let F be a field of characteristic
0 and let S1,S2,...,S r be all p-regular sections of G.
Proof.
Then
Apply Theorem 3.14 and Lemma 3,121ii) together with the fact that
Si I-+ Sii’is a permutation of the set ISl,. ..,S r 1 . 4. CONLON’S THEOREM, FONG‘S DIMENSION FORMULA ZLND RELATED RESULTS
Our main goal in this section is to investigate the decomposition of induced modules from normal subgroups and the dimensions of projective covers of irreducible FGmodules
.
study of S ( F G ) .
As a preparation for the proof of the main theorems, we prove a
We do this for use in the sequel where we will pursue our
number of important results which will be frequently applied at a later stage. Throughout, G denotes a finite group, field. Let
N
a normal subgroup of
G and F a
A l l modules are assumed to be finitely generated.
V be an FN-module and let g E G.
whose underlying space is
V
We define
to be the FN-module
and on which the elements n E N act according to
the rule
We say that two FN-modules V 1
g E G.
and
V 2 are C-oonjugate if
V
’V
for some
It is clear that G-conjugacy is an equivalence relation, so the set of
FN-modules is a disjoint-unionof G-conjugacy classes.
A straightforward
CONLON'S THEOREM
verification shows that V
is irreducible (completely reducible, indecomposable)
.
if and only if ' Y
is irreducible (completely reducible, indecomposable) consider the set H = I g E GIV
Given an FN-module V , subgroup of
G containing N.
be an irreducible FN-submodule of
U and the
VN.
Let H
Then there exists a positive integer e
$7 (i) VN cz e ( 'U @ for H
in G
... 8 g n U ) , 91
and
In particular, VN (ii) W
gn
U,,..,
W
denote, respectively, the
VN
isomorphic to
U.
such that
,...,gn}
is a left transversal
is completely reducible.
is an irreducible FH-module such that
(Here, of course, eU Proof.
V g W
and
e
means a direct sum of
First of all, it is clear that
module isomorphic to g U .
V =
Therefore each gU
is a sum of irreducible FN-modules gU. that "CyV)
Setting
W. = 2
Y
C
wV
for all x,y E G
gU, 1 g i
G
copies of
U).
C gU and that gU is an FNSfG is an irreducible FN-module. Thus
It follows, from the definition of
and so
X
U
yU if and only if zH= yH.
n , we deduce from Proposition 1.3.6, that
9fgp
VN
=
FJ
@
... CB wn
(W
=
and that
W.ze
z
gi i U
for some positive integer e i' Fix g E G , i E {1,2 and
V;
U are pairwise nonisomorphic irreducible FN-modules.
N
gV,
and
11 = g , , g 2
where
W "eU
VN
is a
be an irreducible FG-module and let U
of all submodules of
sum
Then H
is G-invariant.
V
Let V
4.1. THEOREM (Clifford(l937)).
gV}.
2
as the i n e r t i a g r o u p of
We shall refer to H
in case H = G we shall say that
inertia group of
141
s E H.
,...,n } ,
and write ggi = 9 . e 3
Then
gWi = g C g.hU =
%Hz
C g.8hU = W %H3 j
for some j E 11
,...,n}
W)
CHAPTER I11
142
and obviously z(yVi) = (q,')V the set { V action.
,. . .,Wn 1
i'
for all z,y E G .
transitively and H
Thus the group G
is the stabilizer of
W under this
It follows that dimWi = dimW and hence that all the ei
F Invoking Lemma 2.4.1(ii), Finally, because
v
acts on
are equal.
F
we also deduce that
W is an FH-module and V
1
8.
is irreducible, so is W and the result follows.
In order to make further progress, we need to introduce the notions of projective representations and twisted group algebras. Let F*
denote the multiplicative group of F and let Z2(G,F*) be the set
of all functions
which satisfy the following identities
Z2(G,F*)
We shall refer to the elements of define aB
as c o c y d e s .
Given
U,8 E
Z2(G,F*),
by the rule
It is then obvious that aB
is also a cocycle and that Z ' ( G , F * )
becomes an
abelian group. Let t : G
* F* be such that
t ( l )= 1
and let &t: G x G
-+
F*
be
defined by
We shall refer to B2CG,F*1
bt
as a cobounduq.
It is routine to verify that the set
of a l l coboundaries constitutes a subgroup of
Z2tG,F*).
The factor
group
H* (G,F*)
=
z 2 ( G , F * ) / B ~LG,F*)
is said to be the second cohomoZogy group of
G
over F*.
The elements of
CONLON ' S THEOREM
are called cohomology classes; any two cocycles contained in the same
H2CG,F*l
Given a E Z2(G,F*), we shall
cohomology class are said to be cohornologous. write
143
a
for the cohomology class containing a.
One can easily check that if the requirement (1) is dropped, then the resulting cohomology group will be isomoiphic to the one introduced. Let
V be a vector space over F.
projective representation of G
A
mapping
over F
p :G
d GL(V)
is called a
if there exists a mapping a
:G x G -
F*
such that
Thus an ordinary representation is a projective representation with a(Z,y) = 1 for all z,y E G.
If we identify G L ( Y )
with
GL(n,F),n = dimY, the resulting F map is called a projective matrix representation of G over F . As in the case of ordinary representations, we shall treat the terms "projective representation" and "projective matrix representation" as interchangeable. In view of the associativity of the multiplication in G, conditions (3) and (4) imply that a E Z2(G,F*).
shall often refer to TWO
p
To stress the dependence of
p
as an a-representation on the space
projective representations pi : G
+
GL(Yi),i = 1.2
V
on
and a ,
we
v. are said to be
linearly equivalent if there exists a vector space isomorphism f
:
V1-
V
such
that
for all g E G.
The projective representation
p
on the space
v
is called
irreducible if 0 and V are the only subspaces of V which are sent into themselves by all the transformations p(g),g be two a-representations.
defined by
Then the map
€
G.
Let pi : G -+
GL(Yi),i = 1,2,
CHAPTER I11
144
We warn the reader that unlike
is easily seen to be again an a-representation.
the situation for ordinary representations, there is no natrual way to define a sum of an a-representation and %-representation when a # B. p
on the space
An a-representation
V is completely reducible if for any subspace W invariant
under all transformations P ( g ) ,g E G, there exists another such subspace W' with
V = W 8 IJ'. over F can be interpreted as FG-modules.
G
Ordinary representations of
The same situation prevails for projective representations in which the role of the group algebra FG plays the twisted group algebra defined below. a Given a E Z2CG,F*), denote by F G the vector space over F with basis which is in one-to-one correspondence with G .
E GI
PG
in
Define multiplication
distributively using for all x , y E G r,-
xy
=
a&,y)G
a Then F G becomes an F-algebra to which we refer as the twisted group algebra G
over F.
Note that if a & , y ) = 1 for all z , y E G, then FaG
FG.
More
generally, we have:
4.2. LEMMA. Proof.
FG
P G
if and only if a a
Assume that F G say f
G ' F into F, so does ,
and so a
=
gt, where t
assume that a = 6t J, :
FG.
:
G
--t
for some t
is a coboundary.
Since FG admits an F-algebra homomorphism
FaG 4 F .
:
F* :
G
Then, for all x , y E G,
is defined by +
F*
P G + FG, which is the extension of
with +
t(g)
=
f(g).
tC1) = 1.
Conversely,
Then the map
t ( g ) g by F-linearity gives the
desired isomorphism. m
4.3. LEMMA.
Let A
unit group of A
be an F-algebra and let f
which satisfies
be a mapping of G
into the
145
CONLON'S THEOREM
Then the mapping f*
FaG
:
4
A
defined by
of F-algebras. Proof.
Since
p (LA
-
f* is the extension of g
vector space homomorphism.
To prove that f*
99
) =
f(g)
I-+
CXgf(g,
is a homomorphism
by F-linearity, f *
is a
preserves multiplication, it
suffices to check it on the basis elements g,g E G.
Since
the result follows. The next result shows that the study of a-representations is equivalent to the
a
study of F G-modules.
4.4. PROPOSITION. of G
There is a bijective correspondence between a-representations
a
and F G-modules.
This correspondence preserves sums and maps bijectively
linearly equivalent (irreducible, completely reducible) representations into isomorphic (Irreducible, completely reducible) modules. Proof.
Let
p
be an a-representation of
G on the space V.
Due to
Lemma 4.3, we can define a homomorphism f : FaG + EndCV) by setting and extending by linearity.
Thus V
F
a
becomes an F G-module by setting
(Xg
(CXgslu = EX p(g)u 9 a Conversely, given an F G-module define
so that
pCg1 =
p
f(5).
Then
V, and hence a homomorphism f
p(g) E
GLIV) since
is an a-representation on
dence between a-representations and A subspace
V.
9
V.
E
F.g
E G,u E V )
FaG --t. End(v), F is a unit of P G . Moreover, :
This sets up a bijective correspon-
FaG-modules.
W of V is invariant under all p(g)
PGsubmodule of
f(i) = p(g)
if and only i f
W is an
Therefore the correspondence preserves sums and maps bi-
jectively irreducible (completely reducible) representations into irreducible lcompletely reducible) modules.
CHAPTER 111
146
We next observe that an F-isomorphism f a
F G-isomorphism if and only if now that
=
if(V)
pi : G + GL(V .) , d = 1,2,
linearly equivalent to such that p 2 ( g ) f
p2(g)f(V) = fpl
=
(g)V
p2
:
a
Vl-
V2
of F G-modules is an
for all g E
fC$)
E Vl.
G,V
Assume
Then P1
are two a-representations.
is
if and only if there is an F-isomorphism f : V l +
fpl(g),
for all g E G.
or to gf
The latter is equivalent to for a l l
( V ) = f(sU),
V
.
g E G,V E V
Hence two
a-representations are linearly equivalent if and only if the corresponding modules
'
are isomorphic.
Given a E Z2(G/N,F*) , we denote by
infa the element of
Z2(G,F*) defined
by
a k infa induces a homomorphism H 2 (G/N,F*) -+
Then the map
the i n f k t i o n map.
FBG by n H
embed FN in 4.5.
n,
THEOREM CCliffordC1937)).
for all
v
Let
Then there exists w
FN-module.
V
Note that since (infcl)(x,y) TZ
E N,
=
H 2 (G,F*), called
1 for all x , y E N , where
B
= infa
we may
or B
=
be an absolutely irreducible G-invariant
= UG(V)
E Z2(G/N,F*)
such that for y = info,
can be extended to an FYG-module, i.e. there exists an FYG-module W
that WN
2
V.
Furthermore, if w
(infa)-'.
is a coboundary, then
V
such
can be extended to
an FG-module. Proof.
Let
I? be the representation of N afforded by V .
To prove the
first statement, it suffices b y Proposition 4.4) to exhibit y E Z2(G,F*), whose values are constant on the cosets of N , that PCn) =
r(n)
for all n E N .
a nonsingular matrix LCgl
By hypothesis, for each g E G
of
G such
there exists
such that
We may, of course, assume that L(1) = 1.
G containing 1 and set
and a Y-representation p
Let T
be a transversal for N
in
'
CONLON S THEOREM
147
Then, for all g E G , n E N ,
From Schur's lemma, we readily verify that a scalar
p ( x ) p ( y ) differs from
p ( q )
only by
:
with y ( x , y ) E F * .
Thus p
is a Y-representation of
Furthermore, since L ( 1 ) = 1, we have Given x,y E G
and n ,n 1
2
E N,
G
and hence Y E Z 2 ( G , F * ) .
p ( n ) = r ( n ) for all n E N .
we have
whence yLml , y n 2 ) = Y ( Z , Y ) . By the foregoing, we may write that w
t
:
G / N A F*
with p*
for all n E N .
tU )= 1.
(V)
E
inf CW) =
for some
W
E Z2(G/N,F*).
t(xN)t(yN)t(xyN)-'
Then, setting
P*(g) =
is an ordinary representation of
G
Assume
for some
tCgN)-lp(g),
it is
such that p * ( n ) = p ( n ) = r ( n )
This completes the proof of the theorem.
In view of Theorem 4.5,
G
=
is a coboundary, say W ( x i V , y N )
immediate that
W
y
Z*(G/N,F*).
the extendibility of
V
is governed by
For this reason we shall refer to
wG(V)
as the obstruction
t o the extension of V to FG. 4.6.
LEMMA.
Let A
be a finite-dimensional F-algebra, let V
be an absolutely
irreducible A-module, and let W = U @ V, where U is some finite-dimensional F For each a E A, let @aE EndV be defined by @ u ( v ) = u v , vector space over F .
v E
v.
(i) If
F 8 E EndW is such that for all a E A, 8(1 @ @a) = (1 @ F
$,)€I,
then
148
CHAPTER I11
8 = $
for some $ E EndU
81
F (ii) If 6 E G L ( W )
8‘l(1 @
= 1 @ ‘C-’@~T,
then 0 =
,...,um 1
Ci) Let { u
Proof.
are such that for all a E A ,
T E GLCV)
and
+8T
+
for some
E
GL(U).
Then, for all v E V,
be an F-basis for U.
we have
rn =
e(U.82)) 2
for some 0 . . E EndV.
”
a E A,
For a l l
c ~ . Q B3% ..(v) j=l 3
(7)
we have
F
m
Because the
{u.) 3
are linearly independent, we deduce that
or, equivalently, that 8 have
Ji
E EndV
A
g j z . = h ..*lV for some h . . 32
Then $ E G L ( U )
3%
and, by (71,
(it) Put y = 1 8 T.
. F.
Since
V
i s
absolutely irreducible, we
Define $ E EndU by
F
8 = $ 63 1, proving (i).
Then
and thus, by (i), 6Y-l =
II, 8 1
for some
JI E G L ( U 1 .
Hence
8= ($@l)y=+8
‘I
as we wished to show. Suppose that the map
defined by
p
i
: C .-,GLCV.) 2
,i = 1,2, is an a+-representation. Consider
149
CONLON'S THEOREM
p1 @ p 2
Then
is obviously an a a -representation.
We shall refer to
1 2
@
as the (inner) tansor product of the projective representations P I
and
In module-theoretic language, the (inner) tensor product can be defined Let V and W be FO"G and FBG-modules, respectively.
V
@
F"BG-module where
W is an
as
p2
pz.
follows.
Then the vector space
the action of the elements i,g E G, is defined by
F
and then extended to
V @ W and F '@G
by F-linearity.
F Let a E Z2CG/N,F*) an Finf(')G-mcdule
g,g
and let
V be an PCG/N)-module.
inflV) whose underlying space is V
4.7. THEOREM. CCliffordU937)).
as being infZated from
Let H
be the inertia group of V
tion to the extension of to FinfwH
(by Theorem 4.5.
V.
8, let w = w H ( I ' ) E Z2(H/N,F*)
V
is G-invariant.
and
T :G
=
infw,y = inf(w -+
-1
)
V
Then there exists an
U such that
By Theorem 4,1Ci), MN
that for all n E N,w E W, and v E
B
denote any extension of
such extension always exists).
we may therefore assume that M
put
be an obstruc-
Owing to Theorem 4.1Lii1, we may harmlessly assume that H = G
Proof. which case
be an irreducible constituent of MN.
to FH and let ext(V)
P-l IH/Nl -module
irreducible
V.
Let F be an algebraically closed field, let
be an irreducible FG-module and let V
of
on which the elements
G, act according to the rule:
E
In the future we shall refer to inf(V)
M
Then one can form
and S
=
=
in
is a direct sum of copies
W 8 V, where W is an F-space such
V,
ext(V).
If rl
:
N+
GL(V),p
GLG) are the respresentations afforded by V , M ,
:
G-
and S ,
GL(M)
C W T E R IIL
150
r e s p e c t i v e l y , then
and t h e r e f o r e
Thus, by Lemma 4 . 6 C i i )
, for
= qCg)w,W E W,
Setting
each
g E G
i t follows t h a t
M Furthermore, because
M
$(n) = 1
W
GL(W) such t h a t
i s an F%-module
such t h a t
W 8 extV
_y
W
is i r r e d u c i b l e ,
taking i n t o account t h a t
$Cg)
there exists
for all
must also be i r r e d u c i b l e .
n E N,
we deduce t h a t
Finally,
W
2
inf(U)
-1 f o r some i r r e d u c i b l e
CH/N)-module
U.
T h i s completes t h e proof of t h e
theorem. I n o r d e r t o apply Theorem 4.7, we need t h e following simple observation
4.8. LEMMA. order:
be an a l g e b r a i c a l l y closed f i e l d and l e t
G
be a group of
n.
If
(ki)
CY E
Z2(G,F*)
and
group a l l elements of which have o r d e r d i v i d i n g
rn is t h e order of t h e cohomology c l a s s of
01,
is cohomologous t o a cocycle of o r d e r m.
Ciikl
If
charF = p
Proof. CY E
F
H2(G,F*) i s a f i n i t e
(i)
a
Let
Z2(G,F*).
0,
then
p does not d i v i d e t h e order of
We f i r s t note t h a t (iii)is a consequence of C i i ) . Because
aLz,y)a(q,z) = a(y,z)a(t,ya),
we have
H 2 (G,F*). Assume t h a t
n. then
CONLON'S THEOREM
Setting t ( y ) =
151
it now follows that
6 G
Thus an = A t ,
proving that each element of
To prove that H2CG,F*)
H2CG,F*) has order dividing
is finite, it suffices to verify (ii).
71.
Indeed, if (ii)
is true, then H2(G,F*) can be regarded as a subset of all mappings from G to the group of n-th roots of H'(G,F*)
in F.
1
G
But the latter is finite, hence
is finite. a(z,y)" = t ( t C ) t ( y ) t ( x y ) - ' , x , y E G,
To prove (ii), write Because F
is algebraically closed, for any
that p ( r ) "
= t(3cI-l.
up=bp implies u=b.
for some t and hence
:
G+
c1 =
4.9. LEMMA.
3
=
a(6p),
3c
E G
0.
for some t
there exists
Ll(3c)
: G--,
E F*
F*.
such
we deduce that 8" = 1, as required.
(a-b)*
Then, for all a,b E F,
0.
Suppose now that
F*.
6p,
8
Setting
Assume that charF = p
= Up-bp,
so
E Z2(G,F*) is such that
Then a(z,y)P = p
~ p(y)p p p(xy)-P for some p
: ~4F*
as required. be an algebraically closed field of characteristic p > 0
Let F
and let a E Z2(G,F*). (i) G
X
Then G'F
2
FG under either of the following hypotheses:
is a p-group
(ii) For any prime
q # p,
the Sylow q-subgroups of
G are cyclic or general-
ized quaternion (Liil G
is a cyclic extension of a p-group
Proof. (if1 Let q
Ci)
Direct consequence of Lemmas 4.2 and 4.8.
be a prime and let S be a Sylow q-subgroup of
standard fact of cohomology theory Ce.g, see M Hall(1959)) of H 2 (G,F*)
is isomorphic to a subgroup of H 2 C S , F * ) .
quaternion,then H2CS,F*I = 1 we may assume that G
(see Karpilovsky (1985)1 .
is a cyclic q-group and q # p.
G.
It is a
that the q-component If S is generalized Hence, by Lemma 4.8,
Let g be a generator of
152
CHAPTER I11
G, say of order m. with pm
so that
=
G ' F F-basis of .
-m 9 = h * i far some h
Then
=
()I;)"
Thus FO"G
i.
F*.
E
Then the elements
Choose any
? . E l
- m-1 l,Llg, ..., (Vg)
-
c
F
form an
FG.
(iii) Direct consequence of (ii). 4.10. LEMMA. Let F be an algebraically closed field such that charF
IGI
and
c1
Then dimV divides the order of G. F By making a diagonal change of basis E G I of PG, we may
let V be an irreducible F G-module. Proof.
by any cohomologous cocycle.
replace CX
is of finite order, say m.
that a
F and let G* f(Eii)
=
E
=<Ez;lg
Thus, by Lemma 4.8 (ii), we may assume
Let
G, 1 G i G m>.
be a primitive m-th root of
E
Then the map f
:
G*+
G
1 in
defined by
g is a surjective homomorphism whose kernel i s a central subgroup
Furthermore, if then p* : G
p :G
GLCV)
-+
+
GL(V)
is an +representation of G
defined by
irreducible representation of
p*CEzgl
G*.
=
Eip(g)
Since charF
k
<E>.
afforded by ,'l
is easily seen to be an IG*l
G*FE> ' G ,
and
the
result follows by appealing to the following standard fact (see Curtis and Reiner (1981)1 :
If F
is algebraically closed, charF
1 IGl
and
Z is a central subgroup of
G, then the dimensions of irreducible FG-modules divide Let F be an algebraically closed field.
4.11. LEMMA.
G'F
i s a local ring if and only if
Proof.
Assume that G
Lemmas 4.2 and 4.8, p"lG Conversely, assume that
=
PH
3
0 and G
i s a p-group. 7
0.
Then, by
FG and hence, by Corollary 1.2, F"1G is a local ring.
a
P G is a local ring and assume that
G
has an element
It follows from Lemma 4.9 that for H=,PH
Hence
F~HFx Fx proving that
p
Then, for any G # 1,
is a p-group and that charF = p
g of prime order p # charF. is semisimple.
charF
(G:Z).
... x F
has nontrivial idempotents.
P G , a contradiction.
(p times)
But then the same is true of
So the Lemma i s proved.
We are now ready to prove the following important result.
EFfl
CONLBNG'S THEOREM
153
Let
N
be a normal subgroup of
closed f i e l d and l e t
V
be an i r r e d u c i b l e FG-module.
4.12.
THEOREM.
(G:N),
assume t h a t
G/N
i s p-solvable.
G,
Then
If
dimV
F
be an a l g e b r a i c a l l y
charF = p > 0
(G:N)d,
divides
F
is t h e dimension of an i r r e d u c i b l e c o n s t i t u e n t of Proof.
let
divides
d
where
vN-
W e f i r s t show t h a t t h e r e s u l t i s t r u e under e i t h e r of t h e following
hypotheses:
y
(i) charF
(G:N)
charF = p > 0 and
lii)
G/N
i s a p-group
W be an i r r e d u c i b l e c o n s t i t u e n t of
Let
W,
and l e t
W E
VN,
H
let
be t h e i n e r t i a group of
Z2(H/N,F*) be an o b s t r u c t i o n t o t h e extension of
W
FH.
to
-1
By Theorem 4.7,
t h e r e e x i s t s an i r r e d u c i b l e
d i d = dim i n f (5') F F
Since
?
S
U€/N)-module
such t h a t
dimW = dim extlW1 , we have F F
and
dimV = (dims) (dimW) (G:H) F F F I n case l i )
d i d d i v i d e s (EI:N1, by Lemma 4.9, so dimV F -1 F i s a l o c a l r i n g , by Lemma 4.11. In case (ii), F@ W/N)
therefore
dimV
1G:N)dimW F Turning t o t h e g e n e r a l c a s e , we u s e induction on
r e s u l t i s t r u e f o r groups of lower o r d e r than charF = p
thesis. containing
Since
3
0 divides G/N
CG:N),
G.
\GI.
(G:N)dimW. F dimS = 1 and F
So assume t h a t t h e
By t h e above, w e may assume
i n which case
G/N
is p s o l v a b l e , by hypo-
i s p s o l v a b l e t h e r e e x i s t s a proper normal subgroup
N and such t h a t
i r r e d u c i b l e c o n s t i t u e n t of Then
Hence
.
again divides
F
that
divides
G/M VM
is either a
and l e t
p o r p'-group.
w
be an
S be an i r r e d u c i b l e c o n s t i t u e n t of
S is obviously an i r r e d u c i b l e c o n s t i t u e n t of dimV F
divides
dimW
divides
( G : M ) dimW F
and by induction hypothesis
F
Let
M of
(M:N) dimS F
vN.
By t h e foregoing,
wN.
G
CHAPTER 111
154
Hence d i d divides
F
(G:M) (M:N)dimS = (G:N)dimS F F
as asserted. 4.13. COROLLARY. (Dade(l968),Swan(1963)).
Let F be an algebraically closed
field of characteristic p > 0
be an abelian normal subgroup of a
p-solvable group G. Proof.
and let A
The the dimensions of irreducible FG-modules divide
(G:A).
Direct consequence of Theorem 4.12 and the fact that dimensions of
1.
all irreducible FA-modules are equal to
We next concentrate on decomposition of induced modules from normal subgroups.
4.14. THEOREM(Ward(1968),Willems(1976)). be an indecomposable FN-module, and let H
Let F
be an arbitrary field, let
be the inertia group of
V.
v
Suppose
that
where the ti)
Vi are indecomposable FH-modules.
fi fl @ fl @ .., @ f , =
that L$
2
(ii) If
1
J$
2
implies
Vi
where the
Then are indecomposable FG-modules such
V j'
Vi is irreducible for some i, then I$
Proof. (i) The equality
fl = fl @ ... @ f
induction is transitive and preserves direct sums.
and
Ir are irreducible.
follows from the fact that To prove that each
{
is
n
indecomposable, assume that V? = X 8 Y is a direct decomposition. is identifiable with a direct summand of Schmidt theorem, that Vi
form h @
V
with
t#)N
h E H.
'i
we may assume, by the K r u l l -
is a direct summand of XHI say
XH Next we observe that
I$)H,
Because
=
vi @ X'
(9)
is the direct sum of isomorphic FN-modules of the Applying the Krull-Schmidt theorem, we obtain from (8)
CONLON'S THEOREM
155
Thus, for all g E G, there i s an FN-isomorphism
for some positive integer mi,
g 0 Vi
I
mi(g Q V)
(1 ,G
i ,G
s)
which in turn yields
where
T is a transversal for H in G containing 1.
Restricting (9) to N ,
However X
we see by (10) that
is an FG-module, so
proving that for all
t
T , t 8 V appears as a direct summand of XN with
E
multiplicity at least mi.
t
@
V and
t 28
Taking into account that for distinct t ,t 1
2
E
T,
G V are nonisomorphic FN-modules, we infer from Cii) that (Vi)N
is isomorphic to a direct summand of
XN.
Thus
ViG
d i m 8 G dimX and therefore
FZ
F
is indeed indecomposable. We are left to prove that contradiction that direct summand of
d i m E
F and hence that L$ = W
.
F"
is irreducible.
Next we exhibit a relationship between Idempotents In a ring R, and idem-
patents in a factor ring R/I, where I is a nil ideal. denotes the image of P E R 4.15. LEMMA.
R'
p
= R/I.
I be a nil ideal of a ring R.
Let
(I) Each fdempotent
-e =
in
In what follows
E
can be lifted to an idempotent e E 8 ,
E
that is,
E.
(ii) If e
and f
and only if
eii
(iii) Let I= potents in
R'.
fR as right R-modules if
then eR
are idempotents of R,
as right R-modules. E
+
... +
En
be a decomposition of
-1
into orthogonal idem-
...,e
Then there exist orthogonal idempotents e l , e 2 ,
E
R
=
E.
such
that 1 = e
Moreover,
+
... + en
i is primitive if and only if ei
E
Proof. (i) Given an idempotent u-u2 E
and
I and hence
( U - U ~= ~0 ~
E
of
E,
for some m
ei
= E
i
is also. choose u E R
a 1.
We have
with
Then
157
CONLON'S THEOREM
rn
Note that on the right each term after the first rn while the first rn the first rn u(1-u) E
terms, then 1
-
=
-
=
u =
ba = f.
to f.
and f
=
denotes the
sum
is a polynomial in u .
of
Now
a,b E R :
eR-
2
Thus e
fR
=
(e-'e)e
where z E e r e .
Because
' = zz ' = z ' z .
a h (e-z") = e .
Next write b a 2
2
a
f-y,
=
y
2
YE
ZE
a ,b
=
1
(f-y)2
and since y
Because a 2 b 2 = e
e,
=
and
b
=
=
e-l(fi)
b, =
to b
fbe
so
=
b,
e - l ( f ) b = a b , and
satisfy the condition above, then
with inverse y -
ay.
satisfy conditions equivalent to this
and a , b E R
2
=
=
and b a 2
are such that a Then a b
e a f , b1 = fbe.
1
eaf, 1
= e-z
,
=
Z+Z"
= ZZ"
=
Z"Z
and hence
b (e-z"), then
then y E fIf..
f-y = Hence y 2
e-l@)
Setting z" = ez ' e , we obtain
=
e a f , b = j%e,ab
=
i s nilpotent, 1-2 has an inverse 1-2' so that
2
NOW put
1
to f R
fbod I ) , and set a
e , ba
a
satisfy the same conditions for the isomorphism ZH'FR,
Now suppose that
fbe,ab
and
=
fR
We first show that eR
fR is an isomorphism that maps e
and let a,b E R
z,z
Then
is a required idempotent.
b and @ ( e ) e= @ ( e l , hence be
=
a.
such that
is a homomorphism from eR
FE.
u (mod I),
are idempotents of R.
Conversely, if a
ba.
isomorphism.
1
fb
Then
Assume that eR
z+z
2 m ~ ~ ~ l ( l +- u )
Assume first that 0
similarly f
E
where g
m
and similarly eaf =
b
Hence, if e
Because e ( 1 - e ) = e ( 1 - u ) g = 0 , e
E.
if and only if there exist
so
.
...
+
u2m
(ii) Assume that e
bx
rn e+(l-u) g ,
,
I, so
that is, e
3~ t+
u
terms are divisible by
e
and a
rn
is dfvisfble By (1-u)
1
(b a
Since
2
f2-fy-yf+y2
=
2
)'
=
b ea 2
2
=
b a
2 2
,
we have
f-2y+y2
is nilpotent, we find that y = 0 and so b 2 a 2 = f. 2
=
f
imply a
= ea
f , b 2 = fb,e,
the desired
assertion is proved. (iii) Assume that e
and f
are idempotents of R
first show that there exists an idempotent g
such that
such that
=
7
z F = F z = 0.
We
and eg
0.
=
ge
=
158
CWTER 1x1
To t h i s end, note t h a t
h2 = h,he = 0 ,
Then
is n i l p o t e n t and so
fe
and
7.
=
is a u n i t .
1-fe
g = (1-e)h.
Now p u t
Then
proving t h e a u x i l i a r y a s s e r t i o n . W e now show t h a t i f
0, G group, and c
the sum of all p-elements of
G
(including 1).
Our
a finite
aim is to
discover necessary and sufficient conditions under which
In particular, it will be shown that J(FG) Given an FG-module
V, we write
xy
=
Ilk?) in case G is p-solvable.
for the character of
G afforded by
V
and
set
If
H is a subgroup of G, then lH denotes the trivial FH-module.
Define the
CHAPTER XI1
172
F-linear map
T
FG
F C
Ezgtf
t
and put KerT Finally, if
= It E
V
5.1. LEMMA. (i) KerT
(ii) If S E Syl (GI
FG.
=
k(c)
and
W.
V
T can be brought into argument.
=
(lS)
G , then Xv(xl
have TIDI
c
t
=
Kerx
for all g E G
g E FG, then tr(g-'r)
(Lemma 3.12Ci) 1.
= D
Z
=
XY
Hence, if g
Hence T(gcc)
E
G.
g
for all g E G Since G-l
P
= tP(g(zC1)
=
and
G
P'
we
and so T ( g x ) = o
Y be left transversals for N(S) in G and S in N(S), G G
respectively, and for a given g E G ,
Then
for all
if and only if zc = 0.
Let X and
CiiI
SIT(z)
V.
0 if and only if tr(gz) = 0 for all g = tP(3CC)
:
G
(i) If 2 =
D =
is an ideal of FG.
IIv(S)
=
SfG
hence
for the
The following simple obser-
and, in particular, KerT
In particular KerT
Proof.
0 for all g E GI
as a composition factor of
vation illustrates how the map
P
=
is irreducible, we write n(V,W)
V
V,W are FG-modules and
multiplicity of
D E
FGITCgcc)
set
is a left transversal for S
in G
and
is not a p-element, then by (1) XV(g)
Let g be a p-element.
Then g E P
=
T(g) = 0 .
if and only if g
NCP), for all G
P E Syl (G), and therefore M = {P E Syl (GI I# = PI. We claim that P P if sustained, it will follow from (1) that X ( g ) = IrJ(S): S I T ( g ) , l(modp); IMl V G
A CHP\RACTERIZATION OF ELEMENTS OF
T(g) = 1,
since
and hence t h e r e s u l t .
(GI I
lSyl
1
P
5.2.
[MI
< g > a c t s on S y l ( G ) P
d u c i b l e FG-module.
is a splitting field for
F
by
Hence
'
(modp) , as required.
Assume t h a t
LEMMA.
173
i s e x a c t l y t h e number of o r b i t s of l e n g t h 1.
lMl
conjugation and
The p-group
J(FG)
FG
V
and l e t
be an i r r e -
Then
KerXv = ann(V)
Proof.
rX ( V )
n
Let
= dimV
x E FG,
and f o r each
let
F
End(V1 with M n ( F ) , F homomorphism FG 3 I V ~ ( F ) of F-algebras given by x = 33,z) E
V.
Identifying
Tr(r
only v e r i f y t h a t
r = o,
r
then
X
X
rx
=
r r
=
) =
=
9 2 g E G.
0 for a l l
=
0 for all
and l e t
V 1
,...,V
J ( F G ) = Kerx Proof.
1 < i
Then
A E Mn(F1.
Assume t h a t
F
Put
x E
n
KerXi,
Tr(r
Kerx
=
0.
gx
Tr(r r ) 9 x
Hence we need
rX
k
n(Vi,V)
for a l l
and denote by
Then, by t h e d e f i n i t i o n of
then f o r a l l
3
Conversely, assume
0.
If
g E G
and hence
Since t h e l a t t e r i s p o s s i b l e o n l y i n t h e c a s e
is a s p l i t t i n g f i e l d f o r
= n(V$,V)
V -
=
Conversely, assume t h a t
0 for a l l
=
xi X
g E G,
Phi
proving t h a t
Tx.
i f and only i f
and we claim t h a t
If
-
FG, l e t
V
be a l l nonisomorphic i r r e d u c i b l e FG-modules.
ri
s.
G
E End(V) be defined by " F we o b t a i n a s u r j e c t i v e
'
i f and only i f p
V
g
0 and hence
0, t h e r e s u l t follows.
5 . 3 . LEMMA.
Vi,
gx
0 for a l l
TP(r ) = gx
Tr(A*r 1
g+
r
n KerXi x E KeryV.
Then
V'
be an FG-module Then
i E 11,...,s } .
t h e c h a r a c t e r of
G
afforded by
CHAPTER I11
174
C r .x .(gx1 2 2 PXri
Let E
= e
1
1
+
... ,E
J(FG),
=
es
+
for all g E G
= 0
J(FG) be all block idempotents of FG/J(FG)
Vi, if necessary, we may assume that
By renumbering the
Vi
identity transformation on
(3)
and annihilated all V j
Ei
.
acts as the
with j # i.
Hence, for
x E FG,
any
if i = j (4)
if
z
Now (3) implies that
r.X.(e .(gr))= 0, for a l l
g E G.
i # j
Invoking ( 4 ) , we
vkr.
I .I
deduce that X i ( g s ) = 0
gor all g E G
n KerXi, proving
xE
and all i
such that p
k
ri.
Hence
(2).
P k i Applying (2) and Lemma 5.2, we derive
S
Since J(FG1 = n ann(Vi)
and since for any proper subset X
of
{ V l,...,vsl
i=1
J(FG)
C
n ann(V1, the result follows.
V€X 5.4.
LEMMA.
Let F be a splitting field for FG, let V be an irreducible FG-
module and let S E Syl (GI.
P
(i) n(V,FG) = dimP(V)
F
G
(ii) dimP(V1 = ISln(V,(lS1 1 F Proof. (i) Let e be a primitive idempotent of FG such that
V
1
FGe/J(FG)e.
Theorem 3.6(ii),
Then, by Proposition 1.10.16 (iii), n(V,FG) = dimeFG. But, by F Since P t V ) FGe, (i) is established. dimeFG = dimFGe.
F (iil is
Since lS
IS!.
F
is the only irreducible FS-module, the composition length of FS
Let
O = X C X 0
be a composition series of FS. we have
1
C . . . C X
Is1
=FS
Then, by Lemma 2.4.2(11 and Proposition 2.4.3(1),
J(FG)
A CHARRCTERIZATION OF ELEMENTS OF
0
=
XoG C X G C
... C X G I S I =
(FS1 G
2
175
FG,
where, by Proposition 2.4.4(iii1,
Invoking (i), we therefore derive
as asserted. As
a final step in the preparation for the proof of the main result, we
establish the following property due to Okuyama (1980). 5.5.
Let F
LEMMA.
r potent of FG sections of Proof.
t r ( e z ) = 0.
and let Z
G.
=
P'
C FS;
=
,
where S1,
... ,Sr
be a primitive idemare all p-regular
ez = 0 if and only if t r c e z ) = 0 . P' ' 0, then clearly t r ( e z ) = 0. Conversely, assume that
Then, for any Z
If ez
let e
be a splitting field for FG,
E
Z
Given y E FG, we have e y = eye
+ (eey - eye)
and so
ezFG C e z F G e + [FG,PG] Because F is a splitting field for G, e F G e / e J ( F G ) e
eFGe
=
Fe
+
2
F
and so
eJ (FG) e
Invoking Theorem 3.14, it follows that
ezFG C - Fez and hence that t r ( e z F G 1
=
+ e J ( F G ) z e + [FG,FGl = F e z + IFG,FGl 0.
Thus
ez = 0 as asserted.
We have now come to the demonstration for which this section has been developed. 5 . 6 . THEOREM (Brockhaus (19851).
Vl,...,V
r
Let F
be a splitting field for FG, let
be all nonisomorphic irreducible FG-modules and let S E Syl ( G ) .
Then the following conditions are equivalent: (i) J(FG1 = L(c1
P
CHAPTER I11
176
(iil
SocFG
(iiil p
j'
FGc dimP ( Vi)
=
for a l l
iE
{l,.
.. ,r}
Is1 (iv) cP(vi) # (v) cP 0.
and kernel N ti)
If p
If p
IN1
divides
P is a Sylow p-subgroup of N
and
(and hence of
G,
then J(FG1 = FG-ICPI
by Lemma 7.l(iii)),
(iil
be a Frobenius group with complement H
does not divide
I N ! , then
J(FG) = J ( F H ) e and dimJ(FG) F
=
dimJ(FflH)
F
where
Proof.
(il
Assume that p
By Lemma 7.1 (ivl, N
of N .
divides
IN\ and that P
is a Sylow p-subgroup
is nilpotent and hence P 4 N .
P 4 G and
Thus
the result follows by virtue of Proposition l.l(iiil. (ii)
Suppose that p
that F
does not divide
is algebraically closed.
Hence FN(1-el
Note that FN = FNe @3 FN(1-el
=
0
be a field of characteristic p P
and kernel Iv.
4
(FN (1-el 1 G
and therefore J(FG) = J(FGe1.
7.8. COROLLARY (Wallace (195811.
Then
FNe
1
1N'
Invoking
we deduce that
FGC1-el
J V G ] (1-el
and
is a direct sum of nontrivial irreducible FN-modules.
Lemma 7.6 and Proposition 2.4.3(11,
Thus
By Corollary 1.18, we may assume
Let P and l e t
is completely reducible
Now apply Lemma 7.3.
be a Sylow p-subgroup o f
G,
let F
G be a Frobenius group with complement
CHAPTER I11
190
where
In particular, dimJ(FG) = IPI
-1
F Proof.
Apply Theorem 7.7 for H = P
and Corollary 1.2.
We close this section by considering the following two examples. 7.9. EXAMPLE.
p = 2.
Let C = S
Then G
G =,
and let
is a Frobenius group with complement P = < b > and kernel
N =. The element e in this case is given by e
1 -(l+a+a2) 3
=
=
(l+a+a2).
Since I(P) = { A (b+l)I A E F) , Corollary 7.8 implies that J ( F G ) = {hCl+a+a2)(l+b) Ih E F)
Ix E
= {X(1+u+a2+b+ab+a2bI
Let G
7.10. EXAMPLE.
cbc-l
=
=
G = and let p = 3.
Then G
P = < c > and kernel N =.
=
e3
=
l,ub=ba,eae-l=b,
is a Probenius group with complement
The element e
1
b2
F}
e = -(l+a+b+ab) 4
=
in this case is given by
l+a+b+ab
Hence, by Corollary 7.8,
7.11. REMARK.
Let G be a Frobenius group with complement P E Syl ( G I .
P
According to Burnside (1955,p.335), Hence, by Zassenhaus C1949,p.119), (i) If p # 2 , (ii) If
p = 2,
then P then P
P
has no noncyclic subgroup of order p2.
we have
is cyclic
is either cyclic or a generalized quaternion group.
'
UPPER AND LOWER BOUNDS FOR dimJ(FG)
191
F 8 . UPPER AND LOWER BOUNDS FOR
dimJ(FG)
F Throughout this section, G denotes a finite group, F an arbitrary field of characteristic p > 0, lG or
lFG
the trivial FG-module and u = dimP(lG)
F Our aim is to provide an upper and lower bounds for dimJ(FG1.
It turns out that
F is of maximal F-dimension if and only if the Sylow p-subgroup P
J(FG1
normal, while S(FG1
is of minimal F-dimension if and only if P
of
G is
is disjoint.
Some of the results presented here were originally stated
from its conjugates. for splitting fields.
The following simple observation will eliminate that
assumption. 8.1. LEMMA.
Proof.
Let L be a field extension of
Put
V
=
1FG
.
Then V
Proposition 1.19 and Corollary 1.18. 8.2.
LEMMA.
Let P
= lFL
L
F.
Then
and the result follows by virtue of
H
be a Sylowp-subgroup of
G
and let V be a projective FG-
IPI divides did. F Proof. By Corollary 1.2, PP is a local ring.
module.
module,
Then
V p is a projective module.
8 . 3 . LEMMA.
Let V
V.
is a free FP-
NOW apply Lemma 4.18.
Then P(V1
be an irreducible FG-module.
a direct summand of P(lG) 8
Since FG
is isomorphic to
In particular,
F dimP(V1
u dimV
F Proof.
BY Proposition 2.5.7,
F
P(1 1 8 V
is a projective FG-module.
On the
G F other hand, by Theorem 3.8(1111,
1 c P(lG). G-
Hence
n
...,Vn.
for some irreducible FG-modules V1,
It follows that
V is isomorphic
to
CHaPTER ILI
192
a submodule of
n
n
P(Vi)) = Q soc(P(Vi)) i=1 i= 1
SOC( @
But, Theorem 3.8(iii),
i
SocP(V.)
E c1,. . .,n}, and thus P ( V )
2
2
for all 2.
V.
P(VJ
Hence
V.
V
is a direct summand of
for some
P(lG) 8 V
F
.
We are now ready to prove the following important result. U
be a group of order p rn
Let G
8.4. THEOREM.
(i) (Wallace (1958))
dimJ(FG)
F
(p,rn) = 1.
where
Then
> pa - 1
(ii) (Brauer and Nesbitt (1941)) dimS(FG) 4 I G (
M
-
F Proof. e
,...,e21
... @ FGe,
FG = FGe @
(i) Write
of PG with FGe /J(FG)e
5 J(FG),
Since J(FG)el
Then, by Lemma 8.2,
= dimFGel - 1
dimJ (FGIe
F
lG.
2
1
for some primitive idempotents
'
> pa - 1
F
the required assertion follows.
(iil By Lemma 8.1 and Corollary 1.18, we may assume that F is algebraically closed.
Let V ,V , . . . l V 2
-
be all nonisomorphic irreducible FG-modules.
n
Then
n dimFG/J(FG)
=
F =
C (dimVi) i=1 F
(by Corollary 1 10.17)
p i=l(dimV.
Then
r
By ( i l ,
9-10. LEMMA.
r e g u l a r class of given
E
C , K E CR(G)
i s t h e c o e f f i c i e n t of
3:
contained i n
,
put
S,
Fix
1.
G
Co
and l e t
=
s E Co, P E Syl (C (s1)
P
G
C F C R (Gl
Then
X(C,KI
i s an i n t e g e r such t h a t
I C l p , a KCs
Proof.
Let
co
E C E
CR(G)
.
=
X(C,K)*lF
Invoking Lemma 9.9 ( i i l
On t h e o t h e r hand, a n easy c a l c u l a t i o n shows t h a t
‘1
Since f o r
be a unique
and w r i t e
K+S+
P
t h e r e q u i r e d a s s e r t i o n follows.
be a p-regular s e c t i o n of
G
G
i n t h e r i g h t hand s i d e of
and so t h e above c o e f f i c i e n t is
P E Syl (C (g1 1 ,gy E S P G
S
Let
X
l b o d pl
X
g E C and y
any
r
c (x
, we
derive
and, f o r
’ p-
CHAPTER I11
214
Because the number above is an integer and conclude that h ( C , K )
as required.
lSyl I C (s) 1
P G
I
1 (mod p) , we
is an integer such that
.
We have now come to the demonstration for which this section has been developed.
The following result for the case D 4 G
9.11. THEOREM (Kulshammer (1984b)). of
Let
is due to Robinson (1983).
c1 ,C2 ,...,Crn
be all p-regular classes
G having the same defect group D, let eD be the sum of all block idem-
potents of FG with defect group D and let P Define the matrix MD
=
M..)E M 23
m
OF 1
P
be a Sylow p-subgroup of
G.
as follows:
a and pd are the orders of P and D, respectively (by Lemma 9.10
where p
+
applied for S = c,K = C
j
and C = Ci, the number in parenthesis is an integer).
The? dim2 (FG)e 0 / 3 ( 2 (FG)1 eD = rank
F
MD
(By convention, eD = 0 if there are no blocks of FG with defect group D)
Proof. tives in
Then
Let gl,g2,...,gp
G and put
be a full set of (P,P)-double coset representa-
A
CHARACTERIZATION OF
dimJ(Z(FG))
215
F
N i the transpose of No.
and we denote by
11,...,m} and denote by
Fix j E
6(c+) E 3
Lemma 9.7(ii),
rD(G)and, by
+
Applying Lemma 9.10 for S
= c,K = C
j
ci.. 23
6(C:)
Lemma 9.6(i),
C = Ci,
and
3
=
+
Ci
in C.C. 3 a(C+c). Thus 3
the coefficient of
By
we also have
We now claim that the result will follow provided we show that P
(CiIp'aij = Indeed, put M'
W
Is
=
(a..) ,U $3
i
=
the F-linear span o f
linear span that dimV
of
=
F
,um 1 .
{Vl,.,.
rn C a C?
i=l ij - z
{Vl
+ ?,(GI Because
and
c nkinkj k=l FI
r...,t(rn
=
[s(VD(G) + ?D(G)I/?D(G).
+ YD(G)1.
V C_ VDlg(G) and
d i d = rankMr. By ( 6 ) and (7)
(71
-note by
VD(G) n
fDD(G)
By ( 5 ) , V
the F-
= 0,
we also have
F t rankMD = rankM' = rankN$b
Applying Lemma 9.8, we conclude that
t
dim2 (FGIeD/J (2 ( F G ) e D ) = rankNDND = rankMD
as claimed. By the foregoing, we are left to verify ( 7 ) .
XP
5 PgP
so that XP
=
YgP for some y E P.
To this end, assume that Then
we see
216
CHAPTER I11
-1
x ax
E gkPgk
-1
( g k P ) P = gkP
( a r b )(Icty) = (a,bl
Note a l s o t h a t
and
y-lby E gkPgk
i f and only i f
y E C(b ) n P n g k P g k - l = C(b) G P Thus t h e s i z e
and hence
k
of t h e o r b i t of
(a,b)
i s given by
-1
(gkPIP = gkP
A
CHARACTERIZATION OF
d i d (Z(FG))
217
F
Thus in this case we have
R
=
1 and
Applying (91, we therefore conclude that
( c ~ I ~ = , k=c~1nkinkj ~ ~ proving (7) and hence the result. I.t is now an easy matter toprovidea characterization of
dimJ(Z(FG1).
F 9.12. COROLLARY.
Let
21
be the number of conjugacy classes of
G and let X
be the set of all nonisomorphic defect groups of p-regular classes of each D E X, let the matrix
MD E M
be defined as in Theorem 9.11.
[F )
m P
dimJ(Z(FG1 I
=
r
-
F Proof.
Our choice of
groups of blocks of FG.
G.
For Then
C rankMD
E X
X ensures that it contains all nonisomorphic defect Hence Z(FG) = @ Z(FG)eD
EX where
eD
is the sum of all block idempotents with defect group
if there are no such idempotents.
D and eD
Accordingly,
Taking dimensions of both sides and applying Theorem 9.11, we derive
r - dimJ(Z (FG)I F
=
C rankMD
,
E X
whence the result. We close this section by providing some applications of Theorem 9.11.
=
0
218
CHAPTER I11
9.13. COROLLARY [Green (19681).
D
Let
t E CG ( D )
Then t h e r e e x i s t s a p-regular element
G
be a d e f e c t group of a block of
FG.
Q
and a Sylow p-subgroup
of
such t h a t
Proof.
We keep t h e n o t a t i o n of Theorem 9.11.
t
i t follows from ( 8 ) t h a t rankNDND# 0.
FG,
a block of
regular c l a s s
C
G
of
D
Taking i n t o account t h a t
is a d e f e c t group of
P
Setting
t
.
z-ldx
=
a s asserted.
n
Q = x-lPx,
and
d E gP, we have
Since
P G
dpd-l
Hence t h e r e e x i s t s a p-
D, a n element g E G and d E gP n C
with d e f e c t group
P n 9Pg-l E Syl (C (dl 1 .
such t h a t
D i s a d e f e c t group of
Since
= XDX
gPg-’
=
dpd-l.
C, we conclude t h a t f o r some
-1
it follows t h a t
t
x
E G
C (D) and D = Q n t Q t l ,
E
G
I n o r d e r t o provide f u r t h e r information about i n v e s t i g a t e t h e rank of t h e matrix
MD
dimJ(Z(FG1 it is d e s i r a b l e t o F defined i n Theorem 9.11. In general,
t h i s i s a formidable t a s k and no p r e c i s e formula f o r t h e rank of
G has been discovered so f a r .
t h e group s t r u c t u r e of
i n terms of
MD
However, it i s q u i t e
p o s s i b l e t o provide some important u s e f u l information, which w e p r e s e n t below. A conjugacy c l a s s
is s a i d t o be nilpotent i f
G
Let
LEMMA.
rnD
d e f e c t group
D
contained i n
0 ,(GI
D. If
Then
:C
A(Z(FG))
a nilpotent
P
be t h e number of nonnflpotent p-regular c l a s s e s of
and l e t
.tD be
t h e number of conjugacy c l a s s e s of
and have d e f e c t group
RD Proof.
C+ is
Z (FG) .
element of 9.14.
C of
Let
C1,Cp,
...,Crn
D.
G rankMD G
i s semisimple, so
i s a n i l p o t e n t element of
&(c,+)= 0.
which a r e
mD
be a l l p-regular c l a s s e s of
6(C?)
with
Then
[6(VD(G)) + ?D (G)l/iD(G) is t h e F-linear span of
i s n i l p o t e n t , then
G
G
Hence
G
with d e f e c t group
6(Ci) +iD(G),1GiGrn h(Z(FG)).
But
A CHARACPERIZATION OF
dtmd (2(,?GI 1
219
F
and therefore, by Lemma 9.8 and Theorem 9.11, rankMD < m D . Let C ,C 1
,...,CII ,II
2
=
LD, be all conjugacy classes of G contained in For each i E 11 ,... ,R},
(G) and having defect group D.
0
P
Assume that I is conjugate to y Since x , y E 0 ,(G), x-'y
y
i'
is conjugate to y
Thus I y = 1,
is p-regular.
P
j
-1
fix yi E Ci.
and z-lyEPESyl (GI.
P
so
I = y.
This
proves that
Hence mij
=
O for
m.. # 0 as ZZ
so
ID1
-
i#j
=
N~~ m
ICG(yil
Ip.
ii'
1
< i < R,
I s
Hence rankMD 2 R,
the residue (modp) of
as asserted.
We are now ready to provide the following application of Corollary 9.12. 9.15. THEOREM. r r
r
Define the numbers
P,Tl,
and
T
as follows:
G
=
the number of conjugacy classes of
=
the number of nonnilpotent p-regular classes of G
=
the number of conjugacy classes of
1
G contained in 0 ,(GI
P
2
Then r
-
< did(Z(FG1) < r - r 2 .
r '
F Proof. We use the notation of Corollary 9.12 and Lemma 9.14.
It is a
consequence of Lemma 9.14 that C
R
1, where the ei are orthogonal primitive idempotents of Z(FH1. by
yi
the irreducible representation of
Z(FH1 associated with e i and by Fi
the image of Z(FG) under the homomorphism yio7T, 1 < there exist F-isomorphisms Qij
:
F
j
Denote
i G t.
By Lemma 9.20,
Fi which render commutative the following
-+
diagram
Owing to Lema 6.6(ii), the defect groups of each ei
k E {1,2,
...,t1
is such that the defect groups of
Then, by Proposition 2.2.6(ii), respect to FH all the ei Let L
.
Assume that
ek contain P properly.
Yk(AI = 0 where A
It follows that
contain P.
is as in Lemma 6.6 with
(yk-71) ( e ) = 0, which is impossible.
Hence
are in A .
,...,Lr
denote all conjugacy classes of H with defect group P.
Then, by Lemma 9.17(iil, there exist conjugacy classes C 1 defect group P
and with Li = Ci n C ( P ) , 1 G
i G r.
,...,Cr
of
G with
Thus
P
for some
x.. 23
E
F.
Because t > 1, the commutativity of the diagram above yields
dimJ(Z ( F G ) ) F
A CHARACTERIZATION OF
r 0
(e 1 = y
= y 2
1
T( 2
+
Z X..C.l j = l 1-3 3
e y n(
=
225
1
21
r
y (e 1
= 8
C h.C.1 jZ1 3 3
21
I
1
= 1,
Thus Tr(e) i s p r i m i t i v e . f be a block idempotent of
a contradiction.
Let
Step 2.
we are l e f t t o v e r i f y t h a t
L e m 9.18, FG
w i t h d e f e c t group
FH
n(el = f
By
P.
for some block idernpotent
e
of
w i t h d e f e c t group P. f
Owing t o Lemma 6 . 6 ( i i ) ,
Z(FH)
r e p r e s e n t a t i o n of
L
conjugacy c l a s s some
C E Ck(G)
yn
and t h e r e f o r e Let
e
H
of
+
# 0, w e have
.. . + A
Cs
+ (yn) (cs) =
~
P
0
CG(Pl # 0.
c o n t a i n subgroups conjugate t o
contain
i s of t h e form
n ( Z ( F G ) 1.
FG
Y C f , = 1.
Therefore, by S t e p 1,
a s s o c i a t e d with
n(e) = f ,
9.22. L E W .
for
yT
# 0
yon.
Write
.
+C A~ ~ + ~ +c .~. + X~C: ~ (YW ( C i + k ) = 0 f o r a l l
Then
s
f o r some
E 11
,...,t ) .
Because Cs,
Hence t h e d e f e c t groups of
P.
But i f t h e d e f e c t groups of
Thus t h e d e f e c t groups of
n(el
C n C(P)
Hence'
Z(FG).
p r o p e r l y , then by Proposition 2.2.6 and Lemma 6.5,
contradicts
have
1
Ci a s i n Proposition 2.2.6.
k E 11 ,...,~ t ] while ,
e,
P
i s contained i n
denote t h e block idempotent of
1
n(C:)
denote t h e i r r e d u c i b l e
Since by Lemma 9.17 every
i s an i r r e d u c i b l e r e p r e s e n t a t i o n of
e = A C+ with t h e
y
Let
f.
with d e f e c t group
6 (C) = P , A
with
A.
lies in
a s s o c i a t e d with
is p r i m i t i v e .
e
Y(A)
and so of
e
0 which
=
a r e conjugate t o
But then s i n c e
were t o
p.
(Yn) ( e ) = Y ( f ) ,
we
t h u s completing t h e proof.
Let
D be a subgroup of
o r d e r of Sylow p-subgroups of
G.
G
Then
of o r d e r
pd
and l e t
p
U
be t h e
N(D)/D has a normal pcomplement under G
e i t h e r of t h e following hypotheses
(il
d = a-1
and
(ii) d = a-2,
Proof.
G Let
i s t h e smallest prime d i v i s o r of
i s of odd o r d e r and
p
izer i n
G,
then
i s t h e smallest prime d i v i s o r of
P denote a Sylow p-subgroup of
Burnside (Burnside (1955)) a s s e r t s t h a t i f
\GI
P
G.
G.
A c l a s s i c a l theorem of
l i e s i n t h e c e n t r e of i t s normal-
G h a s a normal p-complement.
W e claim t h a t
CHAPTER I11
226
is abelian and q
If P
1
for any prime 9 # p
IAutPl
dividing
IGl,
G has a normal p-complement.
then
Indeed, assume that g E N(P)
(11)
and let < g > = < g
> x . .. x
< g n > be the
G
decomposition of
into direct product of primary components.
Owing to
Burnside's theorem mentioned above, it suffices to verify that g E C ( P ) clearly assume that p
3
1
Since conjugation by
Il.
1
the assumption that q
Autk',
.
We may
G
IAutPI
gi
induces a homomorphism
ensures that the image of gi
is
the identity automorphism, as claimed. Then N ( D ) / D
Suppose that (i) holds.
has a Sylow p-subgroup S
of order
G
p.
Because
IAutSI
=
p-1
and p
follows from (11) that Y C D ) / D
is the smallest prime divisor of
IGI,
it
has a normal p-complement.
G
Then N ( D I / D
Now assume that (iil holds.
has a Sylow p-subgroup S
of
n
CI
order p
p2. By the foregoing, we may assume that IS( = p2 in which case
or
p (p-1)
if
s
(p-1) 2p(p+ll
if
s
Ep2
IAuGI =
Let q
denote a prime divisor of
order, q
7
p+l
ICl
zP
distinct from p.
1
and so in both cases q
IAutSl.
x
E P
Since G
is of odd
Hence, by (ii), N ( D ) / D
has
G
a normal p-complement, as required. We are now ready to provide an important application of Theorem 9.11.
Part
(ii)d of the following result is due to Brauer and Nesbitt (19411, while the rest generalizes theorems due to Gow ( 1 9 7 8 1 , Kawada (1966) and Michler (1972).
9.23. THEOREM. Let F
be a field of characteristic p > 0 which is a splitting
field for Z(FG), let D orders of D
be a p-subgroup of G
and Sylow p-subgroups of
G,
and let pa
respectively.
and pa
denote the
Define the numbers
nD ,mD ,6D , and LD by: nD = the number of blocks of G with defect group D
mD
= the number of nonnilpotent p-regular classes of
G with defect group D
sD = the number of p-regular classes of G with defect group D ED
= the number of conjugacy classes of
G contained in 0 ,(GI
P
and having
227
dimJ(Z (FG))
A CHARACTERIZATION OF
F defect group D.
Then
G nD G mD equality nD =
is true under either of the following hypotheses:
SD
G is p-nilpotent DC(D)/D
is p-nilpotent
G
N ( D ) / D is p-nilpotent G D
G
is a Sylow p-subgroup of
The Sylow p-subgroups of
d = a-1
and p
d = a-2,
G are metacyclic and (p2-l,lGI) = 1
is the smallest prime divisor of
i s of odd order and p
G
IGl
is the smallest prime divisor of
IGI. Proof. (i) We keep the notation of Theorem 9.11.
If e
is a block idem-
potent of FC, then Z(FGIe/J(ZCFG) since F is a splitting field for Z(FG1.
and so, by Theorem 9.11,
nD = rankMD.
e
4
F
Hence
The desired assertion is now a conse-
quence of Lemma 9.14. (ii) If G Lemma 9.22,
is p-nilpotent, then S = D
RD
= mD
and so, by (i), nD =
each of the conditions (f) , (g) implies (c)
group of N ( D ) / D ,
G
By
SD.
.
Since DCCD) is a subG Moreover, (e) implies (a) (see Huppert
(c] implies (bl.
Since (d) obviously implies Ic) , we are left to verify (b).
(1967) ,p.437).
Suppose that D C ( D ) / D
i s p-nilpotent.
If the result is true for N ( D ) ,
G
G the corresponding result is true for G.
then by Lemma 9.17 and Theorem 9.21,
Thus we may harmlessly assume that D 4 G. Let Ci,C z E Ci,y E C
j
be p-regular classes of
j
with x-ly E P E Syl (GI.
and D E Syl ( C ( y ) )
P G
G with defect group D and let Because D 4 G we have
P
D E Syl (Ch)1
P G
so that r , y E C L D ) .
G
Then r - l D
and y D
are p-regular
CHAPTER I11
228
Thus X.y-'D
elements of the p-nilpotent group D C ( D ) /D. However, x-lyD
is also a p-element, so x
It follows that if = xd
with zd = dx
Hence r - l y = 1 and so x
.
conclude that rankM 2 s asserted.
=
.
y.
is a p-regular element.
-1
y E D, say x-ly = d with d E D.
which is only possible in the case d
=
1.
By repeating the argument of Lemma 9.14, we rankMD = nD, so by (i), nD = SD, as
But
10. MORITA'S THEOREM.
denotes a normal subgroup of a finite group G, F
Throughout this section, N
a field of characteristic p > 0 and
Our aim is to provide an important decomposition theorem due to Morita which will be frequently applied in the investigation of J V G )
.
We start by exhibiting some connections between the blocks of FG Assume that f
10.1. LEMMA.
(i) For all g E G, gfg-l ducible representation of
and FN.
is a block idempotent of FN. If y
is a block idempotent of FN.
z (FN)
associated with f, then
is the irreducible representation of
9
is the irre-
defined by
Z (FN) associated with gfg-l.
(ii) The set
is a subgroup of G tiii)
If 2'
containing N.
is a left transversal for G(f)
all distinct conjugates of
f and f*
=
in G,
C tft-l
then the tft-',t
is an idempotent of
E
T , are
Z*(FN)
ET which is independent of the choice of T. (iv) Suppf* Proof. of F N .
is a union of some p-regular classes of
G
contained in N.
(i) Follows from the fact that conjugation by
g is an automorphism
229
MORTTA'S THEOREM
(ii
Straightforward
(iii) It is obvious that the tft-',t E T, are all distinct conjugates of Hence f* is an idempotent of FN g E G
and T
=
1
c
gf*g-l =
T.
which is independent of the choice of
is also a left transversal for GCf,
gT, then T I
c
(gt)f(gt)-l =
f.
in G
If
and
zfz-l = f*
ET
This proves that f* E
ST 1 Z (FG) and hence that f* E Z* ( F N ) .
(iv) Apply Theorem 2.3.4.
.
Let f be a block idempotent of F N . group
covers
G(f)
as the inertia group of f.
f (or B
=
B ( e ) covers
b
=
b Cfl)
In what follows we shall refer to the We say that a block idempotent e E FG
e
if
occurs in the decomposition of
f* into the sum of block idempotents of FG. Observe that conjugation by block idempotents of FN
g E G induces an action of G on the sets of all
and all blocks of F N ,
the elements in the G-orbit of f. G-conjugacy class of f.
and that f* is just the sum of
We shall refer to the G-orbit of
Similarly, the G-orbit of a block b = b ( f )
f as the will be
called the G-cow'ugacy class of b. Assume that b = b ( f ) is a block of FN
Then G(b) = G(f)
and we refer to
G(b)
as
and put
the inertia group of t h e block
b.
(it) The mapping f k f* induces a bijective correspondence
10.2. LEMMA.
Z(FN) and all block
between the G-conjugacy classes of block idempotents of idempotents of
Z*(FN).
(ii) Every block idempotent of FG covers exactly one G-conjugacy class of block idempotents of
FN.
(iii) A block
B = B ( e ) of
FG
covers the block b = b ( f ) of FN
if and only
if ef # 0. Proof.
(i) Let f be a block idempotent of
f* is an idempotent of Z * ( F N ) .
ZCFN)
Assume that f* =
.
UW,
By Lemma 10.1 (iii), where
u
and
V
are
230
CHAPTER I11
nonzero orthogonal idempotents of
Z*(FN).
block idempotents ui,uj
Sfnce
n, 1
i
in F N , 1
u,V E
< j <m,
Z(FN1,
there exist
such that
Thus
m
n c u
f"= Since
c uj i + j=1
is a unique sum of block idempotents of F N
we may assume that u
=
1
a contradiction.
=
zu
=
gZIFN)g-'u
=
Z(FN)u
for a suitable z e Z ( F N ) .
Thus f* is a block idempotent of
Now assume that x
+ f, +
... + f,
But f* = u+u,
Z*(FN).
a left transversal for
for all j E { 1 , 2 ,
...,t}.
the uniqueness of
(*)
Because every g . f g
GCfl
-1
~j
(*)
in G.
Put f
=
f
Then
is a block idempotent of F N ,
implies that
9jf9j for all j E {l,Z,
Then
,
where the fi are uniquely determined block idempotents of F N .
g ,...,gt
so
Z*(FN).
is an arbitrary block idempotent of
x = f
and denote by
.
we therefore deduce that
gfg-l E gZ(FN)ug-' Hence f*
is one of them,
Because u u = u , we have f = u l u E Z ( F N ) U
f.
Bearing in mind that u E Z ( F G ) ,
for all g E G .
and f
...,t}.
-1
E
{fl .,fJ t * .
Hence
x = f*+y for some idempotent y
in
ZCFN)
such that yf* = f*y = 0.
Since x and f*
MORITA'S THEOREM
Z* (FNI,
are block idempotents of
we infer that
Finally, since for any two block idempotents f and f
and only if f
2
Z* (FN).
of FG.
and hence that
and f
of
=
=
f';
z ( F N ) , f*
f*. if
are all block idempotents
k E {1,2,..
. ,n}
such that e
as the sum of block idempotents of FG.
.
if and only if f* = f;
occurs in
This proves
of F N , e
and that, for any block idempotent f
covers f k
x
Owing to (i), we can find block idem-
Thus there exists a unique
the decomposition of f; that e
= 0
of FN such that f ; , c , . . . , f ;
potents f ,f ,...,fn of
y
are G-conjugate, (i) follows.
(ii) Fix a block idempotent e 1
231
Because the latter is equivalent to f
covers f
fk
and
being
Gconjugate, the required assertion follows. (iii) It is clear that ef # 0 if and only if ef* # 0. if and only if e
covers f, the result follows.
Let A
10.3. LEMMA.
be a finite-dimensional local commutative algebra over F .
Then there exists a subfield E
2F
of A
such that
A = E@ J U )
Proof. Observe that if A hence E
3.F .
Since the latter holds
=
(direct sum of F-spaces).
E @ J U ) , then E
A/J(A)
is an F-algebra and
We divide the rest of the proof into two steps.
Step 1. Here we establish the case where J(AI2 = 0 . Put Ap = { $ l a E A }
Therefore, for all a E A , a
apb = 1 for some b E A , A.
A = E @ J(Al,
Ap C - E,
2
If f ( X ) with
2
and
are units of A
then b = ( u - ' ) ~ E A p
Let E denote a maximal subfield of A
E n J U )= 0.
charE = p ,
containing A p .
assume by way of contradiction that
E
and so
2E
E.
Since
(r,p)
we obtain
=
PU
= A/J(A).
is a coefficient of
z.
If
To prove that
a = a + J(A) 4
1, 1 = sr+tp for some t , s E
0.
Clearly
Consider the polynomial f ( X I
Now
2#
is a subfield of
is not irreducible, then it has an irreducible factor because a 4 E.
since J ( A j 2 = 0 .
whenever
and hence A p
the image of E in
We denote by
1 C r < p,
r a E E.
then a 9 J ( A I 2 ,
and note that if up # 0
E.
Since
9-2 E
E[Xl.
(X-aIr
E[Xl
=
(X-a)
E
r
and so
Bearing in mind that
CHAPTER I11
232
a = sra
contrary to our choice of a. is a subfield of
+
Thus
tpa = s(ra) e E ,
f(X) is irreducible, in which case EIal
A properly containing E , a contradiction.
Completion of the proof.
Step 2 .
Assume the contrary and denote by A
a counter-example of minimal F-dimension.
By Step 1, J(A)'
# 0 and so A l = A/J(A)*
Clearly, J(A
J(A)/J(A)'
) =
has smaller dimension than A .
and so J(A1)' = 0.
Because A
is also local,
it follows from Step 1 that
of A l
for a suitable subfield E some F-subalgebra B
Since B
of A .
containing F.
of B
= B/J(AI'
for
Then, by (11, we have
is a local proper F-subalgebra of
subfield E
Write E
A with J(B1
= J(A)
',
there is a
for which B = E ED J(A1'
(3)
Applying ( 2 ) and ( 3 1 , we therefore derive A=E+J(A) and
a contradiction.
10.4. LEMMA. and let y ZCFG)
e
and
Let e and yf
ZLFN).
and f
be block idempotents of FG
and FN, respectively,
denote the corresponding irreducible representations of Then e
covers f
if and only if there exists an F-algebra
isomorphism IT : y
f
U *(FN))
3
ye (2" (FN)1
MORITA‘S THEOREM
233
such that
is algebraically closed, then e
In particular, if F
y (x)
f
Proof.
e
=
e ,e 1
2
Assume that e
,...,er
in FG
covers f .
Jacobson radical of local algebra over F
if and only if
for all x E Z*(FN)
Ye&)
Then there are block idempotents
such that
f*=e and, by Lemma lO.Z(i),
=
covers f
f*
1
+ e
2
+...+
e
r
is a block idempotent of
Z*(F”N).
Then
z*(FN)f*
Z*(FN).
Denote by
J the
is a finite-dimensional commutative
with Jacobson radical J
=
1
Jf*.
Applying Lemma 10.3, we
have
for some subfield E
and a unique z E J
there is a unique y E E
Thus, given x E Z*(FN),
containing F.
Z*(FIVlf*
of
with 1
xf* = y + z Because f*
is the identity element of
Z*(FN)f*
(4)
and since f*e = e ,
from (4) that
xe + ye + ze Since z E J
J(Z(FG11,
w e have
0 =y
e (2)
and thus
=
ye(z)ye(e) = ye(ze)
it follows
CHAPTER I11
234
of ye
NOW the image of the restriction Y ' extension of F
F
of FN,
Hence the restriction y
(Proposition 1.10.8(iv)).
occurs in the decomposition of $*
Of
where
onto y (Z* (FN)1
deduce that
TI =
y6-l
TI
i
E 11
i
,...,k},
Invoking (5) and (61, we therefore
f' is a central fdempotent of FG.
+...+
then y (e.1 = 0 for all i e z
0 = Y , ( f * ) = ny ( f * ) =
f
Hence e
10.5. LEMMA.
covers f
Hence
If e # ei
for all
by Proposition 1.10.8(v).
TIy
f
Owing
ek
are suitable block idempotents of FG.
a contradiction.
is
to
satisfies the hypothesis of the theorem.
p = e 1 + e2 where the e
Yf
is a required isomorphism.
Conversely, suppose that to Lemma lO.l(iii),
Thus
such that
f
x and y are chosen as in (4).
to
e
into the sum of block idempotents
it follows similarly from (4) that the restriction 6
an F-algebra isomorphism from E
of y
onto y e ( Z * ( F N ) ) , by virtue of ( 5 ) .
is an F-algebra isomorphism of E
Because f
is a finite field
to Z*(FN)
(fl =
T(1)
Thus
= 1,
and the result follows. m
Let L be a conjugacy class of N
containing x.
Then
N ( L ) = Ch)N
G Proof. Assume that with
-1 -1
g n mg = n
-1
1
normal.
n E N.
h E NI
xn 1
.
g E NU]. G Hence
Then, for any
"9";'
Conversely, suppose that Then
-1 y
-1
= ng
-1
xgn
G
=x
g E
n
E CG(x) and
E
N, there exists n E
g E CG(")N
since
CGCz)N1 say g = yn with y
and so
-1
-1
g xg = n ma.
E
N
is
C(z) and G
Hence, for all
MORITA'S THEOREM
g E N(L).
proving t h a t
235
So t h e lemma i s t r u e .
G The following r e s u l t i s e x t r a c t e d from a work of Michler(1972). 10.6. PROPOSITION.
,...,e
e
potents
i E { l , ...,s l r Di
(i) For all
i
(ii) For a l l
D
(iv)
N
C
,...,Cr
Civ)).
(i)
yi
Denote by
ei, 1 4 i
s.
Q
k E {l,.. . , r } . f.
1J. (Z* ( F N )
Because
f*
Write
D.
=
y l ( f * l = y Ce ) = 1,
h C+ 1 1
Let
p
GCf)
G
ZCFG) a s s o c i 0 # hi E F
where
contained i n
it follows t h a t
y
i
N
(ck+)
be t h e i r r e d u c i b l e r e p r e s e n t a t i o n of
yi ( Z * (FN1 1
C i E Z*(FN),
yi(Ck+) # 0
# 0 f o r some
Z(FN)
associated from
Ti
such t h a t
it follows t h a t
for a l l
2
71
+
U(Ck)
E 11,
-1
= 1T 7 1 .
1 %
...,s].
+
Yi(Ck)
Thus, by P r o p o s i t i o n 2 . 2 . 6 ( i v ) ,
i s conjugate t o a subgroup of a d e f e c t group, say
D, of
Ck.
Taking
i n t o account t h a t
f* =
and t h a t
Ck
and
(see Lemma 10.1
Then, by Lemma 10.4, t h e r e e x i s t F-algebra isomorphisms onto
t o be a
of p'-index.
... + h,C:
+
D
and choose
1
+
each
ZCFNl
t h e i r r e d u c i b l e r e p r e s e n t a t i o n of
0 # y (Ck) =
and t h u s
f
are s u i t a b l e p-regular c l a s s e s of Because
G(f)
i s conjugate t o a subgroup of
i s a subgroup of
1
with
Dl
Di
DN
Then
be of maximal d e f e c t and l e t
Then
is a s p l i t t i n g f i e l d f o r
GCfI.
Proof. a t e d with
F
e
let
Among t h e block idem-
i s conjugate t o a subgroup of
i s a d e f e c t group of
Assume t h a t
subgroup of
...,s},
E {l,
f,
covering
ei, 1< i Gs.
of
FN.
be a block idempotent of
FG
of
Di be a d e f e c t group
(iii)
f
Let
5 S u ~ p f * ~w e
x 1 c+1
+
... + AP cT+ = e
must have
1
Ck C_ suppej
+
... +
es
f o r some
j E 11,
... , s } .
236
CHAPTER I11
Therefore D is conjugate to a subgroup of D
j
conjugate to D
j-
Since D
and hence, by the above, D
is of maximal order among the D 1
1
,...,D8,
is
we infer
is conjugate to D, proving ti).
that D 1
(ii) Fix x E Ck
together with a complete system g ,...,gn
and choose xi j E CG(x) such that
of the double cosets G ( f ) y C G ( z )
in G
is a right transversal for G ( f )
in G(f)giCG(x).
Assume that g
E
of representatives
N is such that gi-1ggi
=
x for some i E 11,...,nl, and
n
n
c
r.1
-1 is the coefficient of x in f*. Because x E Suppf*, gzw< there is at least one integer t E (1,. ,n) such that p 1 rt. Since and so
2
..
r = t
it follows that G(f)
(iii) Let x
IGlf) I
W gtCG (x)gill
IGCf)
contains a Sylow p-subgroup of
account that g-l xgCSt E Ck, But, by (i), each Di
I G(f)gtCG(x)I - I G
I
Cc(gt-1xgtl.
Taking into
we infer that D is conjugate to a subgroup of C ( f ) .
is conjugate to a subgroup of
be as in (ii) and let L
D, proving (ii).
be the N-conjugacy class containing x.
rn
-1 Then Ck = U ti L t i where { t i l l G i G m3 is a right transversal for NGK) in i=1 G. Denote by l.l the irreducible representation of Z(FN) associated with f.
Owing to Lemma 10.4, there exists an F-algebra isomorphism ylCT),
where T = Z*(FN),
Because
+ yI(Ck) #
such that W(y1 = y (y)
0, it follows that
71
from p ( T )
for all y E T.
onto
237
MORITA' THEOREM
(7)
and hence t h a t
o
p(t-lL+t ) #
4
I t i s obvious that
t-lL+t
normal subgroup of
C
4
f o r some
q
E
11, ... ,m)
(8)
i s a c e n t r a l element of FN whose support i s t h e Nq q conjugacy c l a s s containing t - l x t Since t-lxt E C k , w e may assume t h a t Dl 4 4' 9 4 is a sylow p-subgroup o f c (t-lxt 1 . NOW N n c (t-lxt = c ( t - l t ) is a 4 G 4 4 N q 4 G q
(t-'xt4I,
(t-lxt 1. q 4 containing t-'xtq. 4
group of
L
N
nN
D
Thus
C
f
f
Because t h e d e f e c t groups of
I D 1 n N1 (iv)
D
n N.
f.
Thus
L1
and
gfg-l
i s N-conjugate t o a subgroup of
is t h e sum of t h e conjugates of
l6(fl
Since
For a given
I.
D nN
Hence
f E ZIFN) , we have
g E G , gfg-'
is a Sylow p-sub-
i s a d e f e c t group of t h e N-conjugacy
I t now follows, from (81 and Proposition 2.2.6 ( i v )
of
nN
D
so by Sylow theorem
G q
NOW
, that L
c Ck
5 Suppf*
f o r some
6 (f)
and
f*
g E G.
a r e of t h e same o r d e r , w e have
6(f),
i s N-conjugate t o
N C_ GCf)
a d e f e c t group
1 -
5 Suppgfg-I
class
and so
is a subgroup of
D N
i s a block idempotent of
FN
proving ( i i i ) .
GCf).
whose a s s o c i a t e d i r r e -
d u c i b l e r e p r e s e n t a t i o n i s given by.
Invoking ( 7 ) , we d e r i v e
1:= 1 Assume that
g E G(f).
Z (FN)
s e n t a t i o n s of Let
cosets
Then
{ w j E GI1
G(f)wN(L)
and hence
j in
<s} G.
and
llg
a r e e q u i v a l e n t one dimensional r e p r e s -
p
pg = p.
be a f u l l system of r e p r e s e n t a t i v e s of t h e double Choose
tij E GLf)
such t h a t
{tijUilj=l,...,n.1
G N(LI G
a l e f t transversal f o r
in
i= 1
Then
G
m t.
cu
G(f)wiN(L).
s
'(L+] =
c
ni t . . w s w ' ( L + ) = c nip ;(L+) I: 1-I
i=1 j=1
i=1
,
is
238
CHAPTER I11
t.. w
t..w. p
=
(p
=
)’
wi
for all
? . i
Invoking ( 9 ) , we infer that
E {l,..,,ni}.
J’
and hence that there is an integer d E (1,.
. .,s}
nd.
such that p
Because
it follows that
p ) (G(f) By Lemma 10.5,
:
t G ( f l n iVG(wdLwil))(
NG(L)dLwd-l) = CG(”dxwdl)N,
a Sylow p-subgroup of GCf).
D and D
p-subgroup of
contains
Thus
dividing n.
are conjugate, it follows that D
cG(Wd 3 : W d-1 I .
cG (wd Xw-l)N d
so by (10)
where v ( n ) denotes the highest power of p by ti),
(10)
Because
x E ck and,
is G-conjugate to a Sylow
The conclusion is that
proving (ivl, by applying (11) and (12).
.
At this point we begin to attack Morita’s theorem directly. 10.7. LEMMA. of
Let S
be a ring, let 1 = e
1
+
e
2
+
... + e n
1 into a sum of orthogonal idempotents, and let R = e Se 1
is a subgroup of the unit group of S sitively by conjugation. Proof.
since G
able gi E G , 1
i
Then S
be a decomposition 1
.
Assume
which permutes the set { e l , .
..,e n 1
that G tran-
M ( R ) as rings and R-modules. n
-1 acts transitively, we may write ei = gi e l g i for a suit-
n, with g l
= 1.
Given i,j e 11,...,n3,
set
MORITA’S THEOREM
-1 Then g-’ e , g j # gk e , g k
and fix k # j .
239
so that
( g-1 . e g .) ( g i l e g ) = 3 1 3 i k
o
Thus
and
= e it By definition, eii
ei
=
and therefore
1 = e
11
+
e 22
+
... + Qnn.
The desired
assertion is now a consequence of Proposition 1.5.1. 10.8. LEMMA.
Let
{e
,...,e 1
be a full set of representatives of the G-con-
ei,
be the sum of all G-conjugates of
For each i E {l,.
FN.
jugacy classes o f block idempotents of
let G . = G(e.1
.. , s ) ,
and let ni
let e? 2.
( G : Gi).
=
Then (i) FGe;
p
M, (FGiei) i
(1 Q i
proof. By Lemma 10.1, each e? account that
1
=
e*
... + e*
+
is a central idempotent of
and that e?e? 2 3
=
FG.
0 for i # j ,
Q s)
Taking into
we see that (ii)
is a consequence of (i). To prove (i), fix a block idempotent e
{e = u
1
,...,un
be the G-orbit of e .
also that S = FGe* group of units the set
{*I
and put
By hypothesis,
Because
e* =
G*
=
G(e).
It is obvious that n = (G : G * ) .
is a ring with identity element e * ,
-G = Ge*.
,.. . , V n 1 .
of FN
V
+
-G
and S
Let Note
contains the
acts transitively by conjugation on
... + on
is a decomposition of the
CHAPTER III
240
identity element of S into a sum of orthogonal idempotents of
S, Lemma 10.7
tells us that
FGe* = S Furthermore, since e*e = e ,
Mn(eSel
we have
eSe
=
e (FGe*)e
and it remains to prove that eFGe
=
FG*e.
ege
=
g(g-'eg)e
= 0.
eFGe
is the F-linear span of the set {egelg E G}.
To this end, note that eFGe
If g E G*, then certainly ege
=
=
Hence eFGe
On the other hand, if g 9 G*, then
ge. FG*e
=
and the result follows.
We have now come to the demonstration for which this section has been developed.
The following result is essentially due to Morita (1951) (see also
.
I Nakayama and Shoda (1936), Clifford (1937), and Koshitani (1982~) 10.9. THEOREM.
Let F
p > 0, let N = 0 ,(GI P
be an algebraically closed field of characteristic and let { e
,... ,e
1 be a full set of representatives of
G-conjugacy classes of block idempotents of F N ,
e
where by convention
= 1~l-lc x
3SN
For each i E 11, el
= e?'
... , s } ,
let e? be the sum of all G-conjugates of
let Gi = G[e .I
and let mi = (G
:
Gi).
e
i (so that
Finally, write
for suitable positive integer n,.
for suitable cti E Z2(Gi/N,F*), 1 G i G s . number of conjugacy classes of (ii) If FGez group of
Gi
G
is a block of FG
Furthermore, s
contained in N
and FGe*
for some i E {l,
...,s } ,
coincides with the 2
F(G/IV).
then a Sylow p-sub-
is a defect group of FGe?. Z
(iii) If P E Syl (GI
P
i s normal in
G,
then each FGe?
Z
is a block of FG.
MORITA'S THEOREM
Proof. in
That s
241
1s equal to the number of conjugacy classes of
N is a consequence of Lemma 7.6(1).
G
contained
For the sake of clarity, we divide the
rest of the proof into two steps.
Step 1. Here we prove that if H FG
a central idempotent of
To begin with, put S S
and that S
Then e
e
=
=
2 FHe.
with
and note that e
Let
{ e ..I1 Q i , j
=
e
11
is
n} be a set of matrix units in F H e .
Q
z3
so by Proposition 1.5.1
n
11
e E FH
is the identity element of the ring
FGe = S a M (e S e Since e e
and if
Mn(Fl , then
FHe
FCe
+ e 22 + ... + enn'
11
G
is a normal subgroup of
, we have e 11 S e I 1
=
I1
e (FGe)e 11
) 11
11
e
=
I1
FGe
and thus
11
It therefore suffices to verify that
Owing to Noether-Skolem Theorem (see Reiner (1975)1 ,
for some a E Z 2 ( G / H , F * ) . given g E G , g e l l g Setting u
=
u'
+
is conjugate to e (1-el,
we
there exists a unit u E F H
see that u
unit ut E FH then y
e
such that ug
Let T denote a transversal for H
by an invertible matrix u ' E F H e .
11
-1
11
u = ge
11
g
-1
commutes with
in G ,
such that utt commutes with
e
with y t E F H
=
Hence, for each g E G ,
e 11
and for each 11
.
If y
can be written in the form
y = C y t ET
.
C (ytu-'t)u S!Z'
tt
and hence I1
. t E T choose a
is any element of
FG,
CHAPTER I11
242 -1 e l l (ytut ) e l l E ellFHell
Since
e it follows that e llFGell that e
11
u t # 0
FHe
11
and = e
11
11
(FHe)e
11
For each
FGe
t
E
11
,
(14)
.
Hence
Note
{elluttltE 7'). e
11
u
t have t
{ e u tl t E TI is an F11
t
.
T, set
=
e
11
h E H , t E T.
Then, by the foregoing, e l1 FGell
u t.
t
{sit
associative F-algebra with basis with
I1
is the F-linear span of the set
disjoint supports (lying in the cosets of H t ) 11
Fe
(since utt is a unit) and that the elements
t
basis for e
=
Y by (14), there exists a nonzero CX(z,y1 in F e
Let x , y E T and write
E TI.
(utt)-l(uxx) [u y )
Then
i s an
is a nonzero element in FH
=
ht
and so,
such that
-1
11
((uttl (uxx) (u y 1 ) e = aC;c,y)ell Y 11
But then
--IxyH
xH
= e
11
((utt1-l(uxx) (u y ) ) e Y
=
a(x,y)el,
11
proving (131 and hence the required assertion. S t e p 2.
Completion of the proof.
Observe that the second isomorphism of (i)
i s
a consequence of the first. Apply-
ing Lemma 10.8(11 and Step 1, we have
.
proving (i)
Assume that i E {l,
...,s }
is such that FGe?
Proposition 10.6(ii), there is a p-subgroup Di group of FGe;.
of
is a block of FG.
G ( e . ) which is a defect
Invoking Proposition 10.6(iv), we see that D.
Sylow p-subgroup of
G(eil
,
proving (ii)
Finally, assume that P E Syl ( G )
P
P is a defect group of any block
of
is in fact a
.
is normal in G.
FG.
Then, by
Then, by Corollary 9.13,
Hence, by Theorem 9.23Lii1, the
AN APPLICATION
number of blocks of FG defect group P.
C C_ N.
is equal to the number of p-regular classes of
But a p-regular
C
of
has defect group P
G
The desired conclusion is therefore a consequence of (i).
closed field of characteristic p .
D, then B
Y
If B
Let Di
Gi, 1
i C.S.
2
Since
is a block of
.
and the
i s p-nilpotent
Gi/N
Hence, by Theorem 10.9(i),(iv),
FGe;,
1
< i < s,
be a Sylow p-subgroup
Di and hence, by Theorem
Since FDi is indecomposable, FGe;
FG.
10.11. COROLLARY.
that G
G
FGe? 2 MnSm ( F D i l . z i
lo.S(i) and Lemma 4.9,
true.
.
if and only if
Mn (FD) for some n ? 1.
We keep the notation of Theorem 10.9.
FGe?
with
is a block of FG with defect group
Proof. Q
G
be a p-nilpotent group and let F be an algebraically
Let G
10.10. COROLLARY.
of
243
are all blocks of
Di is a defect group of FG.
So the corollary is
Further to the assumptions and notation of Theorem 10.9, assume
is p-nilpotent and let Di
be a Sylow p-subgroup of
Gi, 1 < i
s.
(i) For any block idempotent e
is the unique block idempotent of i of FN,e? 2
FG which covers ei. (ii) dimJ(FGe?l
F (iiil
’
=
n?m?(lD.I-l) z z
z
did(FG1 = n~m~(IDiI-ll F i=1
Proof. As we have seen in the proof of Corollary 10.10, all blocks of FG and FGe? z is a consequence of (ii).
2
Mnam ( F D i ) .
FGe; ,...,FG*es
This proves (i) and also that (iii)
z i
Since JCFDil
= TCDi),
we have
and hence
as required.
are
.
11. AN APPLICATION : CRITERIA FOR J(FG) = FG*J(Z(FG)) Throughout this section, G denotes a finite group and
F an algebraically
244
CHAPTER I11
closed field of characteristic p
3
0.
Tt is clear that
The problem that motivates this section is to discover necessary and sufficient conditions under which the equality FG.J(Z(FG))
=
J(FG)
holds.
Our main tool
will be a consequence of Morita‘s theorem (Corollary 10.10). Let H
11.1. LEMMA.
G, let R be an arbitrary ring and let
RG( C k ) and, in particular, L ( I ( G ) ) = R ( C g l . M H SEG Proof. Suppose that x = C x g E R ( I ( H ) ) , xg E R. Then x ( k - 1 ) = 0 or gEG = z k , for all k E H . Hence x = z for all g E G,k E H . Therefore
Then k ( l ( H ) )
3:
be a subgroup of
=
gk
9
is constant on the cosets g H
of H .
3:
This proves that
The reverse inclusion being obvious, the result follows. 11.2. LEMMA. If t
Let P be a p-group and let
is the nilpotency index of
R be a field of characteristic p.
JCRP), then J(RPIt-l = R ( E g
By Corollary 1.2, J(RP) = I(P) and so
Proof.
It follows, from Lemma 11.1, that J(RP)
t-1
C R ( C g).
R( C g) is one-dimensional subspace of RG.
&P --
SEP
But J(RPIt-l # 0 and
Hence
sfp
as asserted. 11.3. LEMMA.
Assume that there exists a primitive idempotent u
that all composition factors of FGu Then G
i s
p-nilpotent.
of FG
such
are isomorphic to the trivial FG-module.
9
245
AN RPPLICATION
Proof.
Denote by
lG the trtvtal FG-module.
(ii) and Theorem 3.6(ii), the multiplicity of
Then, by Proposition 1.10.16
lG
FG
as a composition factor of
is dimFGu. On the other hand, by hypothesis, dimFGu is equal to the multiF F plicity of lG as a composition factor of FGu. It follows that if
FG FG
is a decomposition of
i' 2
then none of the FGe
... @ FGe,
@
FGe
into principal indecomposable modules with u G
is linked to none of the e that FGe
=
i G n , has 2
z"
i
lG as a composition factor.
n,
=
e ,
Thus
e
which implies, by Proposition 1.10.14,
is the principal block of FG.
Since lG is the only irreducible
module belonging to FGel, we conclude, from Proposition E.lO(ii),
that G
is
p-nilpotent. 11.4. LEMMA.
then G
Let B
=
B(e1
be the principal block of
FG.
If
is p-nilpotent.
Proof.
Let u
be a primitive idempotent o f
the trivial FG-module. (g-1)u E J (FGI u
FG such that FGu/J(FG)u
Then, for a given g E G , p
and therefore I (GI u
5 J (FG)u.
is trivial, we conclude that I ( G ) u = J C F G I U .
u(mod J(FG)u),
Hence we have
I ( G ) J ( F G )=~ J(FG)~+S, ~ for all n
1
acts trivially on each factor of the series
FGu
3
JLFGIu
3
...
2
so
Since the opposite containment
Recurring this, we get
which shows that G
is
J(FG)'u = 0
CHAPTER I11
246
This implies t h a t each campasitton f a c t o r of
PGu
i s isworpNc
to the t r i v i a l
Now apply Lemma 11.3.
FG-module.
W e are now ready t o prove 11.5. THEOREM (Tshushima (1978b), KIilshammer (1979)1 .
The following c o n d i t i o n s
are equivalent. (i) J(FGl =
FG*J(Z(FC)) B of
(ii) J ( B 1 = B * J ( Z ( B ) ) f o r each block
FG
(iiil
J(B1
(ivl
G i s p-nilpotent with a b e l i a n Sylow p-subgroups. proof.
blocks of
=
B'J(Z(B11
(i)* ( i i ) : Write
FG.
f o r t h e p r i n c i p a l block
FG
= B
@
... @ Bn,
B of
where
El
FG
,...,E
Then
and
Hence
Thus
J ( B . ) = BiJ(Z(Bi)) f o r a l l
(iil * (iiil: (iii)=*
iE
{l,
...,n},
a s required,
Obvious
( i v ) : Write
B = B(el
so t h a t
J(B) = J(FG)e.
Since
are a l l
AN APPLICATION
proving t h a t Let
G
is p - n i l p o t e n t , by applying Lemma 11.4.
G.
be a Sylow p-subgroup of
P
247
B
Then, by Proposition 1 . 2 0 ,
FP
and
thus
To prove t h a t
P
is a b e l i a n , w e use induction on t h e o r d e r of
t h a t f o r any normal subgroup
Let
2
H of
be an element of o r d e r
c p g and
cT=
'I=
P/
of
Let
.
Given
x unless x
f 2.
x
P,
E
in
2
and p u t
c y
is a b e l i a n , by t h e induction hypothesis.
A s s u m e by way of c o n t r a d i c t i o n t h a t
P' = < z >
p
if=
SEP W e may assume t h a t
I t is c l e a r
P.
P, l e t z
be t h e c e n t r e of
P.
w e have
t be t h e nilpotency index
of
is not a b e l i a n .
IxPcP'I= p
J(Z@'P))
Eence
P
and
Then we have
zpr is the conjugacy c l a s s
is t h e F - l i n e a r span of t h e s e t
JVG).
By Lemma 1 1 . 2 , w e have
J(FZltcl
= FT
(3)
W e now claim t h a t it s u f f i c e s t o prove t h a t
Indeed, i f (4) is t r u e , then taking
y E P-2,
w e have
J ( F d = 0
(y-l)T
a contradiction. To prove ( 4 1 , we note t h a t
by v i r t u e of (1).
# 0 and, by ( 3 1 ,
'I
On t h e o t h e r hand,
t-1 E J(FZ) , so
CHAPTER I11
248
,...,Zt - 1 E
for all z
Z,
P-2.
t E
Finally, by definition of
6
(zi-l)
t,
= 0
(7)
i=l where z ,...,zt
are arbitrary elements of
z.
Taking into account (2) and
(5)-(7), we immediately deduce that C4I holds, as required.
(iv) * (i): By Corollary 10.10, for each such that Bi
p-group Di
J(Bi)
2
M
ni
M, (J(FDi)) and Z(Bil
i E {I,. . . , % I ,
(FD.) for some ni
1
z
FDi.
> 1.
there exists an abelian Hence
It follows that J ( B i )
i
GZ
M
( J ( 2 ( B i ) ) and
ni
hence J ( B . ) = BiJ(Z(Bi)), 1 1.
Q
i
n.
Thus
and the result follows. 12. GROUP ALGEBRAS WITH RADICALS OF SQUARE ZERO Throughout this section, G characteristic p > 0 .
denotes a finite group and F
an arbitrary field of
Our aim is to discover the structure of G
in case
J(FG)’ = 0. 12.1. LEMMA.
Assume that the Sylow 2-subgroups of
G
are cyclic.
Then G
is
2-nklpotent. Proof. Let P is a 2-group.
be a Sylow 2-subgroup of
G.
Since P
is cyclic, Aut(P)
Hence the required assertion follows by virtue of property (ii) in
the proof of Lemma 9.22. 12.2. LEMMA. Let B of G
be the principal block of FG.
are defect groups of B .
then J V G )
=
Then the Sylow p-subgroups
In particular, by Theorem 6.4(1),
if J ( B ) = 0
0.
Proof. Let e
be the block idempotent of FG
belonging to B.
We may
RADICALS OF SQUARE ZERO
write
e
2 49
in the form
e for some p-regular classes C 1
module.
Since
V
=
x 1 c+1 +
,...,Cr
belongs to B ,
of
... + xrC+r G.
we have
eu =
U
the trivial FG-
V
Denote by
for all u E V .
Hence for all u E V
which implies that p
ICI
for some p-regular class C
p-subgroup of
G is a defect group of C, as required.
12.3. LEMMA.
Let B
be a block of
Thus a Sylow
8
Then J ( B ) *
FG with J ( B ) # 0.
and only if all irreducible FG-modules in B indecomposable FGmodules in B
in Suppe.
=
0 if
are isomorphic and all projective
have composition length 2.
Proof. Assume that J ( B 1 2 = 0 and write
B = Be @ where
e
1
J(B1 # 0,
> 1.
,...,en
are mutually orthogonal primitive idempotents of
there exists
i
E 11,
...,nl
Since J ( B ) ( J ( B ) e i ) = 0 ,
irreducible and Bei/J(B)ei length 2
... @ Ben
2
such that Bei
... ,e n
As
has composition length
it follows from Theorem 3 . 8 that J ( B ) e i
J(B)ei.
This shows that Bei
and that the composition factors of Bei
idempotents e ,
B,
is
has composition
are isomorphic.
Since the
are linked (Proposition 1.10.14) and Be ./J(B)e .%Be ./JIB) e 2 z 3 j
implies Be. 2 Be (Proposition 1.9.41, the required assertion follows. Conversz j ly, assume that J(B)' # 0. Then J ( B ) * ei # 0 for some i E {l, ...,n] and so
Thus the composition length of Be
i is greater than
2,
as
required.
We are now ready to prove the main result of this section.
.
The equivalence
of (i) and (iii) in the theorem below was established by Wallace (1962a).
12.4. THEOREM. Let F be an arbitrary field of characteristic p > 0 and let G be a group of order p Um
with a
>1
and
( p , m ) = 1.
Then the following
CHAPTER I11
250
conditions a r e equivalent:
o
( i ) J(FGI* =
(iil
J ( B 1 2 = 0,
(iii) p'
p
(ivl
B
i s t h e p r i n c i p a l block of
FG
= 2 =
2
*
G
and
(il
Proof. (ii)
where
has a normal subgroup of odd order and of index
( i i ):
=*
Obvious
p
(iii): By hypothesis,
J ( B ) # 0.
Hence by Lemma 1 2 . 2 ,
d i v i d e s t h e o r d e r of
Since t h e t r i v i a l FG-module belongs t o
But
a p ldimV
(iiiI *
by Lemma 8 . 2 .
Thus
p
G
and so
J(FG) # 0 .
V be a p r o j e c t i v e indecomposible module i n
Let
B.
F (iv)
2
U
= 2
B,
Lemma 12.3 implies t h a t dimV = 2. F a s required.
By hypothesis, t h e Sylow 2-subgroup of
G
i s c y c l i c of o r d e r 2.
Now apply Lemma 12.1.
*
(iv)
(il : By hypothesis,
of order
G
is 2-solvable and t h e Sylow 2-subgroup of
is
G
Hence t h e required a s s e r t i o n follows by v i r t u e of Proposition 1.9..
2.
13. GROUP ALGEBRAS WITH CENTRAL RADICALS Throughout t h i s s e c t i o n ,
p > 0.
characteristic
G
denotes a f i n i t e group and
F
an a r b i t r a r y f i e l d of
Our aim i s t o p r e s e n t S p i e g e l ' s r i n g - t h e o r e t i c proof
(1970) of Wallace's theorem 11962bI which determines t h e s t r u c t u r e of
13.1. LEMMA.
subgroups. Proof.
J(FG)
If
5Z(FG),
In p a r t i c u l a r , Let
K
p
G
then
G
i n case
i s p - n i l p o t e n t with a b e l i a n Sylow p-
( c ' 1.
be t h e a l g e b r a i c c l o s u r e of
F.
Then, by Corollary 1.18,
J(KG) = K 8 J(FG) C - K 8 Z(FG) = Z(KG) F F and so, by Theorem 11.5, t h e l a t t e r implies t h a t
13.2. LEMMA.
Let
equivalent : (i)
XC Z(FG)
X
G
G'
.
is p - n i l p o t e n t with a b e l i a n Sylow p-subgroups.
5 O p t (G) ,
be an i d e a l of
t h e r e s u l t follows. FC.
Then t h e following c o n d i t i o n s a r e
Since
CENTRAL RADICALS
251
(ii) X * I ( G ' ) = 0
X
(iii)
5FGs,
Proof.
*
(i)
a,b E G.
and
z(ab~-~b-~-ll = 0.
Hence
~ba-'b-~,a,bE G ,
X
z E
(ii): L e t
x(ab-ba)= 0.
and
s =
where
Then
Since
G'
i s generated by t h e
t h e required a s s e r t i o n i s a consequence of P r o p o s i t i o n 2.1.10.
( i i l * (iii): D i r e c t consequence of Lemma 11.1 (iii)=) ( i ) : C l e a r l y
{pig E
G)
p
(iil
J(FG)
# FGs,
FGs
i s t h e F - l i n e a r span of t h e s e t
t E G, w e have
5 Z(FG),
then t h e following a r e e q u i v a l e n t :
s =
where
lG'l
Proof.
(i)
s2 = ( G ' ( s = 0 (ii)* (i):
Thus
Since
5 Z(FG).
FGs
If
]LEMMA.
(i) J(FG)
Z(FG).
and s i n c e f o r every
it follows t h a t
13.3.
E
*
(ii):
FGs
and
p
If
J(FG1 # FGS
1
J(FGl
By Lemma 1 3 . 3 ,
5 J(FG).
1,
IG'
Hence
then
e
=
5 FGs.
J ( F G ) = FGS
p
If
I
IG'I,
then
as required
I G r 1-l~i s a nonzero idempotent i n
FGs.
and t h e r e s u l t follows.
The following theorem w a s proved by Wallace (196213) f o r group a l g e b r a s over
I t s g e n e r a l i z a t i o n and t h e proof given h e r e a r e due
a l g e b r a i c a l l y closed f i e l d s . t o Spiegel (1970).
13.4.
THEOREM.
Let
be a f i n i t e group.
F
be an a r b i t r a r y f i e l d of c h a r a c t e r i s t i c
Then
J(FG1
i n g t h r e e types: (i)
G
has o r d e r prime t o
p
5Z V G l
i f and only i f
G
p
and l e t
G
i s one of t h e follow-
CHAPTER I11
252
Ciil
G
abelain
its
P
(iii) I f
is a Sylow p-subgroup of P
complement Proof.
G
G'P
then
i s a Frobenius group with
G'.
and k e r n e l
Let
G,
be a nonabelian group whose o r d e r i s d i v i s i b l e by
p.
It
c l e a r l y s u f f i c e s t o prove t h a t J(FG)
5 Z(FG)
i f and only i f (iii)holds
By Theorem 8.11, t h e l a t t e r i s equivalent t o JCFG) C_ Z(FG) Assume t h a t
i f and only i f
5 Z(FG1
J(FG)
and put
S
dim[J(F(G'P)] = ( P I - 1 F
J(FG) index
c
FGs
(G:N1
and
p )' I G ' 1 .
Clearly
is not d i v i s i b l e by
p.
.
=
G :
and
p
(G'(
(1)
By Lemmas 13.1, 13.2 and 13.3,
rg
N = G'P
i s a normal subgroup of
G
whose
Hence, by Proposition 1 . 8 ( i i ) ,
didCFG) = (G:N)dimJUU) F F Since
JCFG)
C
FGs,
we have
Invoking (2) and (3) w e t h e r e f o r e d e r i v e
did(FN1 C F
on
t h e o t h e r hand, by Theorem 8 . 4 ( i ) ,
a s required.
Then
FNe a FP
Conversely assume t h a t
so
dimJ(FN1 a [PI-1. Thus dimJ(FN) = IPI-1, F F dimCFN) = (PI-1 and t h a t p IG'I. Let F
1
dfmJ(FNe) = [PI-1 = dimJ(FN) F F J(FN) = J(FNe) =
Since that
N
(PI
i s a normal subgroup of
JLFG) = FG*J(FN).
G
J(FN)e
of p'-index.
and thus
J(FN) = J(FNe).
5 FNs proposition 1 . 8 ( i i )
Invoking (41 w e t h e r e f o r e conclude t h a t
Hence
(4) ensures
COMMUTATIVITY OF THE RADICAL
Hence J(FG)
c- Z(FG)
13.5. REMARK.
by Lemma 13.2.
7.11, is in fact cyclic.
.
Thus completes the proof of the theorem.
Let G be a group such that G'P
plement P and kernel G'.
253
Then P
G'P/G'
is a Frobenius group with com-
.
is abelian and hence, by Remark
Furthermore, by Lemma 7.1,
G'
is nilpotent.
14. COMMUTATIVITY OF THE RADICAL OF THE PRINCIPAL BLOCK Throughout this section, G characteristic p > 0, lG
FG.
denotes a finite group, F an arbitrary field of
the trivial FG-module and B,
the principal block of
Our aim is to discover necessary and sufficient conditions for the commu-
.
tativity of J (Bo) Let A
We say that B
block of A .
@ 1
sum
If K
be a
... @ B e n
of principal indecomposable modules, each Be
same multiplicity.
and let B
is quasi-primary if in decomposition B = Be
as a direct
K
be a finite-dimensional algebra over a field
is a splitting field for B ,
i appears with the
then the following
conditions are equivalent: (il B
is quasi-primary
(ii) All irreducible B-modules have the same K-dimension (iii) B / J ( B )
i5 a direct product of isomorphic simple algebras
Indeed, the equivalence of (ii) and (iii) is trivial, while the equivalence of (i) and (ii) is a consequence of Proposition 1.10.16(ii). Recall that if
V
and
W are FG-modules, then the tensor product
v@ w
is
F
an FG-module via
14.1. LEMMA.
Let
cible FG-module.
V
beanFG-module of dimension
Then V @ W
1 and let W
is an irreducible FG-module
F Proof. By hypothesis,
be an irredu-
v = .8'V
for some nonzero V
in
v.
Let X
be a
CHAPTER I11
254
nonzero submodule of
v 8 W.
Then every
x E X can be written as
X = U
8W
F for a w E W.
It is a consequence of the deftnitton of the action of G on
V 8 W that the set F
x is a submodule of W
= { n ) E w ( u 8 w E X 1 # 0
and hence X
x
=
1
Iu
W.
=
Qpwlw E
But
x1
V
has dimension 1, so V 8
=
F
w ,
which proves the lemma. 14.2. LEMMA.
and W be FG-modules with
Let U,V
U
V.
Then as FG-modules
Proof. The canonical map
is surjective F-homomorphism whose kernel is U 8 W.
Since the map obviously
F preserves the action of G, the result follows. 14.3. LEMMA.
Let U be any indecomposable projective FG-module.
sition factors of P ( 1 1
G
tion factors of Proof,
U
If the compo-
have dimension 1, then the dimensions of all composi-
are equal.
Put V = U/J(FG)LI.
summand of P U G ) 8 V .
We
know, from Lemma 8 . 3 , that U
is a direct
Let
F
be a composition series of P ( l G ) .
Owing to Lemmas 12.1 and 12.2,
P ( 1 1 8 V = X 8 V 3 X 8 V 3 .. . > x 8 V = O G F 'F 2 F n F is a composition series of p(l I 8 V
and all composition factors have the same
'F dimension, namely dimV.
F
This proves the required assertion.
COMMUTATIVITY OF THE RADICAL
14.4. LEMMA.
255
The following conditions are equivalent:
(i) All irreducible FG-modules belonging to the same block have the same Fdimension Cii) All irreducible FG-modules belonging to the principal block have F-dimension 1. (iii) All composition factors of P ( 1
have F-dimension 1.
G)
Proof. (iil
(i)
=*
(ii): Obvious
(iii): Direct consequence of the fact that all composition factors of
=*
P(lG) belong to Bo. (iii)
=*
(i):
Apply Lemma 14.3 and the fact that the indecomposable projective
modules in a block are linked. The proof of the implications (il
=*
(ii) and
(ii) * (iii) below is extrac-
ted from a work of Hamernik (1975b). 14.5. PROPOSITION (Morita (1951)).
characteristic p > 0
and let P
Let F
be an algebraically closed field of
be a Sylow p-subgroup of
G.
Then the follow-
ing conditions are equivalent: (i) PO ,(G) 4 G
P (ii) Bo
G/PO ,(GI
is abelian
P
is quasi-primary
(iii) Each block Proof.
G
and
B of FG
(i) * (ii): Set
is quasi-primary
5
=
G/O ,(GI
P
and
=
PO ,(G)/O
P
P
,(G).
Since
is p-solvable (and hence p-constrained), it follows from Proposition 1.20 that
B
FE.
-
By hypothesis, G
1.
Since the same is true for Bo,
the assertion follows.
(ii) * (iii) : Follows from Lemma 14.4. (iii) * (i) : Write B
0
=
B0 (e) and note that, by Proposition 8.1O(i), 0 ,(GI = { g E Glge = el
P
Let
and
Hence FZ has only irreducible representations of dimen-
abelian factor group. sion
-
is a group with normal Sylow p-subgroup P
256
CHAPTER I11
G afforded by the FG-module FGe.
be the matrix representation of
g
if and only if ge
E Kerr
regard
r
as
e,
we have
Kerr = 0
a faithful representation of
G/Op, ( G I .
=
belong to Bo,
factors of FGe
P
that the set of all elements g E G Sylow p-subgroup. say !C
of
33
Moreover, Q
G/Op,CGI.
is abelian.
P
14.6. LEMMA.
Let F
such that J ( B )
Proof.
P
This shows that PO
P'
be a splitting field for FG and let B
(G) 4
G and
be a block of FG
is commutative.
sition factors of Be
If B
is normal in G/O , ( G I
So the proposition is true.
(i) If e is a primitive idempotent of B
(iil
Then it is easily verified
such that r . . ( g ) = 1 for all j forms a
and the corresponding factor group i s abelian. that G/Po '(GI
is quasi-primary implies
Thus we may assume that the entries
are zero if i < j.
(rij(gl)
of the matrix
We may therefore
Since the composition
the assumption that Bo
that all these factors are one-dimensional.
r..Cg)
(GI .
Since
with J ( B ) ' e # 0 ,
then all compo-
are one-dimensional
is the principal block, then B
is quasi-primary
(i) We claim that it suffices to show that dimSoc(Be1
1.
=
Indeed,
F assume that dimSoc(Be) = 1 and let V
be a composition factor of
Be.
Since
F the Cartan matrix of FG is symmetric (Theorem 3.6(vi)), Be/J(B)e tion factor of P ( V ) .
and so PCn
is a composi-
But, by Theorem 3 . 8 ,
If J(BI2P(V) = 0,
has a composition factor of F-dimension 1.
then the composition factors of P ( V l
have the same dimension equal to dimV
.
F Hence dimV = 1.
If J ( B 1 ' E " V )
# 0 then SocP(V)
V
is one-dimensional, by
F hypothesis.
This substantiates our claim.
Since J ( B ) ' e # 0, we have Soc(Be)
5 J ( B ) ' e 5 J ( B ) '.
show that all composition factors of J ( B I '
are one-dimensional.
will follow provided we prove that for all x , y E J ( B )
@a) ( ~ 1 . This is of G
so
since then G'
afforded by J ( B l 2 .
Thus we need only The latter
and U , b E G ,
(ab) (Ccy) =
will be in the kernel of the representation
COMMUTATIVITY OF THE RADICAL
Using the assumption that J ( B I
257
is comutative, we have
as required. (ii) By Lemma 14.4, it suffices to verify that all Composition factors of P(lG) are one-dimensional.
By (I), we may assume that J ( B ) 'P(lG)
composition factors of P(1 1
are isomorphic to
G
=
0.
.
But then all
So the lemma is true.
lG.
We have now come to the demonstration for which this section has been developed. Let F
14.7. THEOREM (Hamernik (1975)).
be an arbitrary field of characteristic
p > 0, let G be a finite group and let Bo be the principal block of FG. Then the following conditions are equivalent: (i) Bo
is commutative
(ii) J ( B o l C_ Z(Bo) (iii) J ( B (iv) G
is commutative
p-nilpotent with a b e Lan Sy )w p-subgroups
is
(i) * (ii): Obvious
Proof. (ii)
=)
(iii): Obvious
(iii) * (iv): Let K
be the algebraic closure of
F and write Bo
= Bo(e)
.
Then
e = e for some block idempotents e lFG,one of the
1 G i G s,
e
i'
say e
1
1
1
,...,e ,
+ e
2
+...+
of KG.
e Since e
acts as identity on
acts as identity on
lKG. Setting
B? = KGe,
it follows that
KQ B F o and that B*
1
= KGe = B*
is the principal block of KG.
... Q3 B: Invoking Corollary 1.18, we derive
CHAPTER I11
258
J ( K Q Bo’ = K Q JWOl = J(B;l
F
... CB J ( B S )
a3
F
which shows that J(B:)
is commutative.
Hence we may harmlessly assume that F
is algebraically closed. By Lemma 14.6, Bo
is quasi-primary and so, by Proposition 14.5,
PO ,(G) d G and G/PO ,(GI is abelian P P where P E Syl (GI.
P
Because G
e
=
is p-solvable, Proposition 1.20 implies that
1 0 P ,K)I-’
c
r
as0 , ( G I
P
Setting H = PO ,(G), it follows from ~emma7.3.
is the block idempotent in Bo.
P
that
on the other hand, by Proposition 1.8Cii.1,
J(FGI
=
FG.J(FH).
Hence
and therefore FG*I(P)e is commutative. Choose an arbitrary s E P
and u E NG(P).
suffices to check that U3u-l = we
d.
By Burnside’s criterion, it
To t h i s end, we put u8u-l
have
Since for any x E FP, ex = 0
implies x = 0, we deduce that
’
(1-s) cl-sl 1 = (1-s)
= 6
E P.
Then
CRITERIA FOR THE COMMUTATIVLTY
OF
259
J(FG1
and hence that s -ss 1
If p # 2
t h i s yields
which case either s
s = s 2 = 1.
=
1
=s-s2
s = s.
Ef p = 2 ,
s
=
or s
then we have s
But if s = s 2 ,
s2.
= s+s2,
in
then s = s 3 and so
and therefore in all cases s
Hence S = 1
+ ss
=
s.
This completes
1
the proof of the theorem. 15. CRITERIA FOR THE COMMUTATIVITY OF J(PG) Throuthout this section, G denotes a finite group and F an arbitrary field of characteristic p > 0.
J(FC1
is commutative.
Our aim is to determine the structure of
G in case
The following lemma is stated in a stronger form than
required. Let FI
15.1. LEMMA.
be a normal p'-subgroup of G.
Then the following condi-
tions are equivalent: (i) J (FG) J (F (G/N) 1 (iil dimJ(FG1
=
dimJ(F(G/N)l
F
F
(iii) Every block
B
V
of
FG with J ( B ) # 0 contains an irreducible FG-module
5 Kerv.
such that N
P E Syl G), PN 4 P
Furthermore, if for
G
then ti), (ii) and (iii) are equi-
valent to each of the following conditions: (iv) J(FG1 = FG'I(P)e, (v)
dimJ(FG1 =
that FGe
1
e
=
1
IN(- E g
1 GI ( /PI -1)
F Proof.
where
SEG
Ipl IN1
Owing to Proposition 2.1.9,
e
is a central idempotent of FG
such
F(G/N).
(i) * (ii1: Obvious (iil * (iii) : Since J ( F G 1 = J(FGe1 @ J(FG(1-e) l dimJ(FGe) = dimJ(FG),
F J(B1
we have
J ( F G ( 1 - e l ) = 0.
and, by hypothesis, Let B
be a block of FC
with
F # 0 and let V be an irreducible FG-module in B .
we have B CFGe and hence eU = U
for all U E V.
Since J(FG(1-el) = 0,
Thus, if
n E N, then since
CHAPTER I11
260
ne = e ,
we have
nu This proves that N (iii) =. Ci):
5 KerV,
Let B
as required.
V and so ev
=
v
FG with J ( B 1 # 0.
be a block of
contains an irreducible FG-module on
for all v E I/
(ne)v = ev = v
=
for all V E
V with N V.
5 KerV.
Thus B C F G e
By hypothesis, B
Then N
acts as identity
and therefore
as required. Now assume that PA7
G.
Q
Then PN/N
and hence, by Proposition l.l(iii)
is a normal Sylow p-subgroup of
G/N
and Corollary 1.4,
and
BY Proposition 2.1.9, of F-algebras.
the map F(G/N1
--+
F G e , g N k ge
Since FG*T(P)e is the image of
determines an isomorphism
F(G/N)T(PN/N) under this map,
we deduce from (1) that
.
Thus, by ( 3 1 , (ivl is equivalent to ti), while by (21, So the lemma is true.
(v) is equivalent to (ii).
In the result below, the equivalence of (il , Cii) and (v) was established by Wallace (1965). 15.2. THEOREM. P E Syl
P
(il
(GI.
JCFG)
Let F
be an arbitrary field of characteristic p # 2
Then the following conditions are equivalent:
5 Z(FG)
(ii) JCFG)
is commutative
and let
(iv) p
)
and J(FG) = F G * I ( P ) e , where
IG'I
(v) The group
G
e
=
261
J(FG1
CRITERIA FOR THE COMMUTATXYITY OF
IGrI-l
is one of the following three types
(a) G has order prime to p (b) G (c)
is abelian
G'P
is a Frobenius group with complement P
Proof.
(i)
and kernel G ' .
* (ii) : Obvious
(ii) =* (iii): Owing to Corollary 1.18, we may harmlessly assume that F is algebraically closed.
Since J(FG)
the principal block of FG. Thus G'
is abelian.
50
P'
is commutative, so is J ( B
and therefore p
(GI
1
Bo
is
is p-nilpotent and P
G
Hence, by Theorem 14.7,
where
)
IG'I.
We now claim that if J ( B ) # 0, then
J(B1'
# 0
(4)
Assume by way of contradiction that J ( B I # 0 but J ( B ) ' = 0 . irreducible FG-module in B ,
then by Lemma 12.3,
If
dimP(V) = ZdimV.
F hand, by Theorem 4.27,
dimP('V)
=
12'1 (dimV)
Hence
On the other
F
IPI
=
2(dimV)
contrary
F p'
F P"
F
V is an
to the assumption that p # 2. With J ( B ) # 0.
N
=
G'
we certainly have PN 4 G.
Then, by (41,
cible module
Let B
be a block of
J ( B ) ' # 0 and so, by Lemma 14.6, B
V of dimension
But then N
1.
5 KerV
FG with
has an irredu-
and so the desired con-
clusion follows by virtue of Lemma 15.1. (iii) * (iv): Direct consequence of Lemma 15.1 (iii) * (v): let M = G'P.
Let G
be a nonabelian group whose order is divisible by
p
By the property (1) in the proof of Theorem 13.4, it suffices to
show that dimJ(FM) = I P I - 1 .
F But M
and
is a normal subgroup of G
of p'-index and so, by Proposition 1.8,
CHAPTER 111
262
as required. (v) =* (i): Direct consequence of Theorem 13.4.
m
The next example illustrates that the hypothesis p # 2 in Theorem 15.2 cannot be ommited without sacrifising the conclusion. 15.3. EXAMPLE (Koshitani (1979)).
Assume that F
p = 2.
Let the group G be defined by
Then G
is 2-nilpotent with 0
P
idempotents of FO
P
(GI
(G) = < a >
and
P
is algebraically closed and
=
ESyl ( G ) .
P
The block
are given by
e = ( l + a + a 2 ) e, = ( ~ + E u + E ~ u )and , e Furthermore b-le b = e
and so G ( e
=
= (~+E~u+Eu)
is of index
2
in G.
Hence,
by Theorem 10.9,
and therefore
The first factor is commutative and the second is also commutative, since
1a2I2= 0. J(FG)
Thus J W G )
is commutative and, by looking at the second factor,
Z(FG).
In order to examine the case p = 2, we need the following preliminary observation. 15.4. LEMMA.
1’
Z( F C ) Proof.
Let I be an ideal of FG.
If I is commutative, then
. Let x,y E 1 and z E FG.
Then
263
CRITERIA FOR THE COMMUTATIVITY OF J ( F G )
so that
commutes with yx and hence 1 '
Z
'
C Z(FG).
Turning our attention to the case p = 2, we now prove Let F be an algebraically closed field of characteristic 2,
15.5. LEMMA. G
P
is odd and a > 1, and let
be a nonabelian group of order 2arn, where rn E
Syl ( G )
and N = G'P.
J(FG)
is commutative
(i1
(ii) G
Then the following conditions are equivalent:
is 2-nilpotent, G'
(iii) G
let
is 2-nilpotent, G'
is of odd order and for each s E G'-1,4
1
ICN(z)I FN
is of odd order and each nonprincipal block of
is of defect 1 or 0. Proof
(i) * (iil:
Since JCFG)
is the principal block of FG. P
Hence, by Theorem 14.7,
In particular, G'
is abelian.
is commutative, so is J(Bol
5 02'(G)
N 4 G and therefore, by Proposition 1.8(i1,
and so
J(FN1
It will next be shown that G' = N'
commutative.
G'
G
where Bo
is 2-nilpotent and
is of odd order.
5 J(FG).
Hence J V N )
Now is
and hence that FN has a
unique (up to isomorphism) irreducible module of dimension 1. Put x
=
C ,g, y SEN
=
Z g,
and z =
.
Since N '
is of odd order, Lemma
SEP
1.3 implies that
whence
' 5 FGZ.
It therefore follows
1 by Theorem 12.4.
Hence, by Lemma 11.2,
On the other hand, by Lemmas 15.4 and 13.2,
J(FG)
that
But I ( P 1 2 # 0 ,
y
E
since otherwise a
I(P)' and therefore yz E FGz.
for all g E G ' .
=
It follows, from Lemma 11.1, that yz(1-g) = O
Since 1 E Suppyx, g E Suppyxg
=
Suppyx C N'P
and thus
264
C W T E R 111
G' C - N'P.
is of odd order, we conclude that N' = G ' .
Because G'
G'-1
Assume by way of contradiction that there exists z has at least two 2-regular class of defect d
Then N 9.23,
FN has at least
principal block B B
2
>
blocks with defect d 2 2.
with defect group D
Mn(FD) for some n b 1.
If n
such that
1, then
=
2.
with
Hence, by Theorem
Thus FN
[Dl > 4.
has a non-
By Corollary10.10,
B has a one-dimensional irreThus n > 1
ducible module, which is impossible by the foregoing discussion. and J I B )
Mn (I(D)) .
4, I(D) # 0
ID1
But since
4 I IcNIZ)1.
and hence J ( B )
is non-
commutative, a contradiction. (ii) * (iii): By hypothesis, each nonidentity 2-regular class of 1 or
0.
G
is of defect
The desired conclusion is therefore a consequence of Theorem 9.23.
(iii) * (i): Let B nilpotent, we have
be a block of
B
1
Since N
Mn (FD1 for some n P 1 (Corollary 10.10).
nonprincipal, then by hypothesis Since N
FN with defect group D.
ID1
is a normal subgroup of G
Mn(I(D) 1
Hence JCB)
2.
of odd index, we have J ( F G I
is 2-
If B =
is
0.
=
FG*J(FN).
Put
Then we have J (FGI = FGJ (FN)e
Since FG*J(FN)el SFGe Lemma 13.2.
1
, FG*J(FNle
Now every block of FN
@
FGJ (FN)e
is a central ideal of contained in FNe
FG, by virtue of
is nonprincipal.
Hence,
2
by the above, S(FNe212 = J(FN12e2 = 0.
and so FG*J(FN)e
is commutative.
Thus
This proves that J ( F G )
is commutative, as
required. We have now accumulated all the information necessary to prove the following result.
J(FGI
CRITERIA FOR THE COMMUTATIVITY OF
G
characteristic 2, l e t
P E Syl (G) and (i) J ( F G 1
(ii1
N
=
a
x
1,G
or
= 1,
x
for a l l
odd, and l e t
s
in
P,
Frobenius group with complement
P/<s>
.
Proof.
l,G
i s 2-nilpotent,
cN ( s l / < S >
a
J ( F G ) * = 0.
= 0,
then
Thus if
commutative.
a
1,
J(FG1
Since
a l s o assume t h a t
a
SVGl = 0
3
then
a
and i f
JVG)'
= 0
#
G'
F
is algebraically
then by Theorem 12.4,
= 1,
and t h e r e f o r e
Hence, by Lemma 15.5,
1.
G ' # 1 i s of
i s e i t h e r a 2-group o r a
i s obviously commutative i n c a s e
1 and
1 i s of odd
G ' # 1 i s of odd
is 2 - n l l p o t e n t ,
By Corollary 1.18, we may harmlessly assume t h a t If
G' #
is 2-nilptentr
a > 1,G
G' = 1, o r
o r d e r and, f o r every i n v o l u t i o n
closed.
rn i s
where
Then t h e following c o n d i t i o r s are e q u i v a l e n t :
G'
1, o r
Q
order and f o r each
(ivl
2 rn,
be an a r b i t r a r y f i e l d of
i s commutative
Either
(iii)
U
be a group of o r d e r
G'P.
F
Let
15.6. THEOREM. (Motose and Nfnomiya (1980)).
265
J(FG1
G'
is
= 1,
w e may are
ii)
( i ) and
equivalent. (i)
*
(iii): A s s u m e t h a t
G'
equivalent,
xyx
-1 -1
y
i s of odd order.
E P n G'
= 1
and so
and hence, by Lemma 15.5, (iii)* ( i v ) :
C (s)/<s>
N
x E
S(FG)
Let
s
Let
x
E G'-1
y E C (z). N
( Pn
P"1
Since ( i l and (ii
i s commutative.
and
Then
This shows t h a t
2.
be an a r b i t r a r y i n v o l u t i o n i n
i s a 2-group.
y E P n ?.
are
Now assume t h a t
P.
C Cs) # P .
N
If
c N ( s )= P ,
then
Then, f o r a l l
Cu(S)-P,
since
IP n
complement
P"l
6 2.
P/< s >.
Hence, by Lemma 7.2,
CN ( s )/<s
i s a Frobenius group with
266
CHAPTER I11
(iv)
*
xE
(ii): L e t
Is1
4 2.
S,
in
G'-1
W e may assume t h a t
S.
# 1, i n which case t h e r e i s an i n v o l u t i o n , say
S
CN ( s ) , and hence
u E N
C N ( s )/ < s
>,
Thus w e have
S C_
9 P" = < s > ,
15.7. COROLLARY.
(Motose and Ninomiya (19801).
G
JVGI
F
Let
S
5'P
.
as required.
be an a r b i t r a r y f i e l d of
be a nonabelian group of order
be commutative.
Now
is a Frobenius group
N
?/<e>.
c h a r a c t e r i s t i c 2, l e t
is not a 2-group. C (s)/<s>
and t h e r e f o r e , by hypothesis,
with complement
and l e t
I t s u f f i c e s t o prove t h a t
Then
which shows t h a t f o r some
S E Syl ICIv(x)).
and l e t
(m,2) = 1, u
2'm,
G
Then t h e Sylow 2-subgroups o f
1,
7
are either
z2 x z
c y c l i c o r isomorphic t o
2
Proof.
Let
of odd o r d e r ,
If
P n P"
x E N-P,
f o r all
P and k e r n e l G'.
Hence
On t h e o t h e r hand, i f
s i n P,
then
N = G'P.
mz-'s-'
P
then
P/<s>.
P n G'
Since
?L
P
either cyclic or
mrcm
= 1.
f o r some Hence
P/<s>
is
x E N-P
cc E C ( s ) N
and some i n v o l u t i o n
-P
C N ( s ) = P.
and so
C (sl/<s> i s a Frobenius group
N
a b e l i a n and i s a Sylow 2-subgroup of
P/<s>
is cyclic.
Thus
P
is
'
OF BLOCKS AND NORMAL SUBGROUPS
characteristic p
B
a-1'
2
Throughout t h i s s e c t i o n ,
those blocks
is
i s a Frobenius group with complement
c N ( s ) / < S > , it follows from Remark 7.11 t h a t
16. THE
G'
i s c y c l i c , by v i r t u e of Remark 13.5.
P n 9 = <s> E
N
Thus, by t h e condition (ivl of Theorem 15.6, with complement
Then, by Theorem 15.6,
i s a b e l i a n and
P
= 1
and l e t
P E Sy12(G)
3
of
normal subgroup of
0.
FG
G.
G
F
denotes a f i n i t e group and
an a r b i t r a r y f i e l d of
Our aim i s t o determine t h e Jacobson r a d i c a l f o r which t h e d e f e c t group of
B
J(B1
of
i s contained i n a given
As a p r e p a r a t i o n f o r t h e proof of t h e main r e s u l t ,
we
THE RADICAl, W BLOCKS
261
shall prove a number of important properties of independent interest. It will be fundamental for future investigations to provide a useful character -rization of defect groups of blocks.
The following preliminary results will
clear our path. Let W
16.1. LEMMA.
be any subgroup of
G
such that W
VH = where the Vi,l
i
is H-projective.
v
...
@
Write
vn
Vi are indecomposable FH-modules, and denote by Qi Then for each i E 11,.
n.
Proof.
Owing to Lemma 2.6.3cii1
16.2. LEMMA.
-1
yi,
Let
,
we
vi
..,nl,
a vertex of
there exists gi E G
i s
i
.
2
conclude that g .Qg-l c Q, as required. t
i
-
W be an indecomposable FGmodule with vertex Q
is a Sylow p-subgroup of
such that
~ x ~ n ~ xprojective ; ~ for some x E G. i there exists .y E H such that x.Qzil n H 2 yiQyil.
By Lemma 2.6.2, each
Setting gi = xi
Q and let H
be an indecomposable FG-module with vertex
G.
Then
(P;&l
divides dimW.
5 P,
where
P
In particular, if
F dimW is prime to p ,
then P
is a vertex of
W.
F Proof. We first consider the case where be an indecomposable FP-module with vertex
F
is algebraically closed.
X and source S.
Since X
Let
V
is a
subnormal subgroup of P, it follows from Theorem 4.23 that S p is indecomposable.
Hence
V
1
S p and so dimV
= (P:X) did
F
F
Now write
where the
Vi are indecomposable FP-modules, and denote by Qi
Then by (11,
1< i dimW.
F
< n.
a
vertex of
‘i (P:Qi) divides dimVi and, by Lemma 16.1, (P:Q) divides (P:Qi), F Thus, for all i, (P:Q) divides dimVi and hence ( P : Q ) divides F
This establishes the case where
F
i s algebraically closed.
CHAPTER 1 x 1
268
Tn general, let E of
W corresponding to the vertex Q. EQ9
W
Then
=
... CD Wm
W @
F where the
P, and let U be a source
be the algebraic closure of
W.z are indecomposable EG-modules.
is a component of E '8 W
and E
W
Since for all $ E 11,.
8=
is a component of E
F
F
F
.. ,m), Wi G
(E @ U) , we F
8 UIG for the EQ-module E @ U. This F F proves that each W. is &-projective and so has vertex Qi 5 Q. By the algeinfer that Wi
i s a component of
braically closed field case,
(P:&.)
and dimW = dimW F E l So the lemma is true. Given subsets X,Y of
(E
(P:Qi) divides dirnWi. E , idimWm, we infer that
..
i
E
In particular, CFG(G1 = Z(FG).
(P:&l
divides each
(P:Ql divides dimW. F
FX the F-linear span of X and by
FG, we denote by
CX ( Y ) the centralizer of Y in X I
Since
1.e.
Let S
5H
be subgroups of
G.
We shall refer
to the map
introduced below, as the PeZative trace map. Let S
16.3. LEMMA.
for S in
H.
5H
be subgroups of
For each z E CFGtS)I
T
G
and let X be a left transversal
put
(z) =
c zzz-1 ECEX
(i) C F G W f CFG(Sl, T H , S ( ~ E) CFG(H)
and T
His
is independent of the choice of
A.
(it1 TH,s
: CFG(S)
CFG(H)
TH,s(CFG(S))
particular I If
-+
D
is a homomorphism of
i s an ideal of
is a subgroup of S ,
CFG (H)-bimodules.
CFGLH).
then for all z E CFGw),
In
269
THE RADICAL OF BLOCKS
(ivl
D
Tf
I (GI
where
Ci1
s E S,
Because any element of
S,
centralize
we have
CFG(H)
FG
5 C(,S).
which c e n t r a l i z e s If
y =
H must a l s o
xs f o r some
3:
E
X and
then
yzy-I = and so
hX
then
is defined i n Lemma 2.2.5.
D
Proof.
G,
is a p-subgroup of
TH,S
(21
b S ) Z (xs1-l =
xszs -1s-1 = xzx-1
X.
is independent of t h e choice of S
i s another l e f t t r a n s v e r s a l f o r
in
H.
Fix
h E H
and note t h a t
Thus
a s asserted. (ii) The map
z E C,,CSI.
T
H,S
obviously preserves a d d l t i o n .
Then, f o r a l l
x
A s i m i l a r argument proves t h a t
X, we have ys
TH,s’zy)
=
= T (z)y
H
Y a l e f t transversal for D i n
(iii) Denote by
transversal for
E
D
in
H.
Fix
y E CFG(HI
and
xy and t h u s
a s required.
s.
Then
XY
is a conjugacy class of
G.
is a l e f t
Thus
as a s s e r t e d .
Uv)
Let
-+
z = C E ID(G) , where
C
Fix
g E C,
CHAPTER I11
2 70
put
Q
Q
L = CG(g) and choose
5 D.
Then
(L1
.
b)
E
cFG(o) and,
L,
G : Q ) is prime t o p ,
by (21,
9.
z = TG,L(gl.
T (L'I C,D
TG,D(L+l =
and so e i t h e r
0
or
E
1,IG).
Co(g)
w e may d e f i n e
K
form an F-basis f o r
C,,(D),
it
But
C.
Thus i n both
'
TG,DIL 1 E ID(G1, a s d e s i r e d . Let
L+
is a d e f e c t group of
+
16.4. LEMMA.
be a subgroup of
G
and l e t
D,H be subgroups of
K.
X a s e t of r e p r e s e n t a t i v e s f o r t h e double c o s e t s DxH i n K k E K put Hk
=
and
kHk-l n D. for all for a l l
D.
Apply-
proving t h a t
TG,D(W) = 2 ,
Since t h e elements
s u f f i c e s t o prove t h a t
f o r each
we must have
L t h e D-conjugacy c l a s s containing g E G and by C t h e G-conjugacy
c l a s s containing
Denote by
z E ID(Gl, w e may assume t h a t
Because
TG,Q'g) i n two d i f f e r e n t ways, we f i n d
T L , & ( g ) = ( L : & ) g and
Denote by
cases
P
Moreover, by t h e d e f i n i t i o n of
ing ( i i i l t o compute
Since
E Syl
a
E
CFG(H)
a E CFG(H) ,b E C,,(D)
Then t h e s e t
Y is obviously a l e f t t r a n s v e r s a l f o r
=
u T(X)X 6 X 8
in
K.
Thus, f o r a l l
a E C,,(H),
we
THE RADICAL OF BLOCKS
271
have
as asserted. (ii) We have (by Lemma 16.3(iii))
(by Lemma 16.3(iii)
)
as required.
.
(iii) Direct consequence of (i) (iv) Follows from (ii). 16.5. LEMMA.
Let A
be a finite-dimensional algebra over F and let e If 11,1 2f...,In are ideals of A
primitive idempotent of A. e E J
+
... + In,
Proof.
1
+ eI
e. 2
with a E Z
a 9 JU). Since A
1
,b
Thus we may assume that A E
I
2'
then not both
is local, a has inverse a',
such that
...,n).
is the identity element of the local ring
implies e E e 1 e 1 = atb
for some i E c1,
Without loss of generality we may assume that n = 2.
primitive, e
If
then e E Ti
be a
eAe
Because e
and e E
is
I1 + I 2
is local and that e = 1.
a and b
lie in J ( A ) ,
which gives
say
1 = a2' E T I ,
as
required. It is now easy to provide a useful characterization of defect groups of blocks. 16.6. PROPOSITION.
Let B = B ( e ) be a block of FG.
Then the following
conditions are equivalent: (i) D is a defect group of B
(iil
D is a minimal element in the set of all subgroups H of G such that
272
CHAPTER 111
B.
Suppose that D is a defect group of
Proof.
Because e
Then, by Lemma 16.3 Civ) , e E IH (GI
e.
assumption that D is a defect group of Assume that H is a subgroup of G
and that
contrary to the
Hence (i) implies (ii).
such that e E TG,H(CFG-(H))
be a minimal element in the set of all such subgroups H 16.4tiv) (with X
ID(G),
Now assume that H c D
Lemma 16.3(iv) implies that e E TG,D(CFG(D)). e E TG,H(CFG(H) 1 -
E
of G.
and let D
Owing to Lemma
G), we have
=
Applying Lemma 16.5, we infer that for some g E G
Then the minimality of
-1
D ensures that gHg
3 D.
Now suppose that H
(and
Then ~ H g - l>_ D and in view of the
hence 9Hg-l) is a defect group of B . implication (i)
* Cii),
16.7. LEMMA. (e.g.
e
we obtain
gHg-' = D, as required.
Let D be a p-subgroup of G
is a block idempotent of FG
(i) For any FG-module (iil
V
If
ti).
If V
satisfies (ii), then eV = V
defined by
O h ) = wx
for all x E eV.
FD and let
with defect
and so (ii) is a consequence of
such that e = TG,D(WI.
all elements g with g E D, 8 E End(VD). in G
B=B(eI
is D-projective.
we may choose w E CFG(Dl
eV
D
is D-projective
To prove (i), we employ the relative trace map.
0 : eV -+
be an idempotent in I (G)
with D as a defect group)
is an indecomposable FG-module in a block
group D, then V Proof.
V, eV
and let e
I E
eV.
Then
Let 7!'
Owing to Lemma 16.3(iv),
Consider the mapping Since w
commutes with
denote a transversal for D
THE RADICAL OF BLOCKS
= e x = x and t h e r e s u l t follows by v i r t u e of Theorem 2 . 5 . 4 .
16.8. LEMMA. If
e.
B
Let
e 9 7,
then
Proof.
Let
=
Ie y
B(el
FG
be a block of
273
. I be an i d e a l of
and l e t
ZCFG) .
i s a n i l p o t e n t i d e a l of
be the i r r e d u c i b l e r e p r e s e n t a t i o n of
Z(FG)
a s s o c i a t e d with
By P r o p o s i t i o n 2.10.8,
We c l a i m that
1 5 Kery;
as required.
Assume by way of c o n t r a d i c t i o n t h a t
is a f i e l d and
Hence
e-x
=
re
i f s u s t a i n e d it wtll follow t h a t
0 # y(1)
y ( e ) = y(xl
Thus
+
is an i d e a l of
f o r some
z(1-e)
x E I
f o r some
y(1l # 0.
y(Z(FG)),
Since
y(Z(FG))
it follows t h a t
and t h e r e f o r e
r
E JCZVG1)
and some
2
E ZCFG).
Conse-
quent 1y ,
e = xe
+
re
and
Now
ZVG).
J(Z(FG))
i s a n i l p o t e n t i d e a l of
a l g e b r a over t h e f i e l d
F
ZVGl
of c h a r a c t e r i s t i c
ZPG)
and
p
3
0.
is a commutative
Hence t h e r e e x i s t s a
CHAPTER I11
274
n such t h a t
positive integer
.
n e = (ze)P
Therefore
16.9.
e
Let
LEMMA.
I,
E
a contradiction.
be a block idempotent of
FG
D
with d e f e c t group
and l e t
A
h
ID(G)
6 ( C ) < D. G i s a n i l p o t e n t i d e a l of Z (FG)
C',
be t h e F-linear span of a l l h
i s an i d e a l of
Proof.
Z (FG)
rD(G) i s
By Lemma 2.2.5,
with
ID(G)
Then
.
T (G) e D
and
C E CRCG),
n
an i d e a l of
Z(FG).
ID,CG)
Since
i s the
h
SUm
of a l l
IH(G)
H c D, we see t h a t
with
T D G ) i s an i d e a l of
.
n
e , e 9 .TDLG),
t h e d e f i n i t i o n of t h e d e f e c t group of p o t e n t i d e a l of 16.10. LEMMA.
E
Z(FG),
e
Let
by v i r t u e of Lemma 16.8.
be a block idempotent of
be any f i e l d extension o f
F
FG
Z(FG1.
By
h
IDb(G)e i s a n i l -
Hence
with d e f e c t group
D,
let
and w r i t e
e = e a s a sum of block idempotents of
EG.
+
1
... + en iE
Then, f o r a l l
{l,
...,n},
ei h a s
D
as a d e f e c t group. A
Proof.
PD(G)
Denote by
respectively, i n
EG.
Fix
and
h
IE(G)
i E {l,
TD(G) and
t h e c o u n t e r p a r t s of
...,nl
ID(G),
and note t h a t , by t h e d e f i n i t i o n of A
t h e d e f e c t group of
e E TD(G)
ei,
it s u f f i c e s t o v e r i f y t h a t
PD(G) and s i n c e PDD(G)
i s an i d e a l of
ei E TECG)-.TZ,(G). 2 (EGI
,
Since
w e have
On t h e o t h e r hand,
A
and so, by Lemma 16.9, A
Let
Z(EG).
Hence i f
A
e E PDCGl,
i
T E ( C ) e i s a n i l p o t e n t i d e a l of
then
ei = eie E T;(G)e,
S be a subring of a r i n g
i f any e x a c t sequence
R.
a contradiction. An R-module
M
So t h e lemma i s t r u e .
i s s a i d t o be S-projective
THE RADICAL
Y
0-
-+
OF
BLOCKS
X-
M-
275
0
of R-modules and R-homomorphisms which splits as an S-sequence, also splits as an R-sequence.
If L C_ S S R
M is ;%projective.
are subrings and
M is L-projective, then obviously
We state without proof the following basic fact (see Higman
C1955, Theorem 4 ) and Hochschild (1956, Proposition 2) 16.11. LEMMA.
Let S
be a subring of a ring R
.
and let
M be an R-module.
Then the following conditions are equivalent: (i)
M is S-projective
(ii) The canonical map
$ :R @ S
M+
i.e. there exists an R-homomorphism (iii) There exists an S-module V
M
given by
q ( 8~m )
=
mn
is a retraction,
R €3 M with $4 = 1. S such that M i s a direct summand of R 8 V.
4
:
M+
S
(iv) Consider any diagram of R-modules and R-homomorphisms
with exact row.
If there is an S-homomorphism
commute, then there is an R-homomorphism
q
:
M+
$
:
M+
X making the diagram
X making the daigram commutep
In order to take advantage of Lemma 16.9, we make the following simple observation
.
16.12. LEMMA.
Assume that the following is a commutative diagram of R-modules
and R-homomorphisms
CHAPTER 111
276
0 If
then the following are equivalent:
= ly,
( i ) Im($XaI
5 Ima
(ii) There is a $ : Z-+ C such that $0 = B$
and
k$
= lZ
(in particular, k
is a retractionl. Proof. Assume that Im($ha)
which shows that %$p = 0 ,
5 Ima.
Because Ima = KerB, we have
by virtue of the fact that U
is an epimorphism.
Hence
and thus $ = B$0-’
i s a well-defined R-homomorphism from
Z to C with the
required properties. Now assume that (it1 holds.
Then
and therefore Im($Xa] SKerB = Ima. The next observation will allow us to take full advantage of the results so far obtained. 16.13. LEMMA. that AR =
Let
s
be a subring of a ring R ,
let A C_S be a subset such
RA and let U be an $-projective R-module.
Then M = U/AU
i s an
277
THE RADICAL OF BLOCKS
S-projective R-module (the fact that AU
U is a consequence
is a submodule of
of the equality RCAU) = ARU = A U ) . Proof.
Consider the diagram below where a l l maps are canonical.
)M i
x
K
Bearing in mind that
we obtain
The desired assertion is now a consequence of Lemma 16.12.
.
We have now accumulated all the information necessary to prove the following
fundamental result. 16.14. THEOREM. (Green (19591, Knorr (1976)). be an arbitrary field of prime characteristic p
Let G be a finite group, let and let B
be a block o f
F
FG.
278
CHAPTER I11
If P
G, then the following conditions are equivalent:
is a p-subgroup of
P is a defect group of B
ti)
All indecomposable FG-modules in B
(iil
ducible FG-module M Proof.
in B
Suppose that
(i)
*
(ii) is true and that (ii) holds.
and there is an irreducible FG-module V is P-projective, Q
M
hand, because P
and
is a vertex of M.
such that P
Then all indecomposable FG-modules in B
a defect group of B .
Since V
are P-projective and there is an irre-
in B
Denote by are &-projective
Q is a vertex of
such that
is conjugate to a subgroup of
P.
On the other
is Q-projective, P is conjugate to a subgroup of
Q are conjugate.
Owing to Lemma 16.7,
then all indecomposable FG-modules in B
Q.
Hence
if P is a defect group of
are P-projective.
v.
B,
Thus we need only
verify the following assertion:
Tf
P
i s a defect group of
B,
such that P i s a v e r t e x of
then there i s an irreducible FGmoduZe
M
in B
M.
(3)
For the sake of clarity, we divide the proof into three steps.
Step 1. Let d
The case where F i s aZgebraicalZy closed.
be the defect of B
By Theorem 2.2.7,
and let p"
be the order of Sylow p-subgroups of G.
there exists an irreducibleFGmodule M
pWd
is the highest power of p
in B
such that
dividing dimM
F
Q a vertex of M.
Denote by
and so Q C P .
is' indecomposable, M
Because M
On the other hand, by Lemma 16.2,
(S:Q)
is P-projective
divides dimV, where
F
G S is a Sylow p-subgroup of G.
d IPI = p , this means that Q = P
Because
and
G thus P is a vertex of
Step 2.
Let
E
M, proving
(3).
be the algebraic closure of F.
every irreducibZe FG-module
M
owl aim i s t o prove that
i s FG-projective.
Owing to Corollary 1.18, FG/J(FG)
is a separable F-algebra and
JBG)
=
E 8 S(FG)
(4)
F Denote by
U
a projective EG-module such that M
U/J(EG)U.
Then, by (41,
279
THE RADICAL OF BLOCKS
J(EG)U
=
and so M
J(FG)U
conclude that M
Invoking Lemma 16.13 with A = S(FG),
U/J(FG)U.
denote the block idempotent of
FG
central idempotent of EG, we may write block idempotents of EG. a defect group of B ' .
V
EG-module
=
Put B'
=
e
0.
Because e
contained in B . =
+
e
... + e n ,
F.
where the
is a
ei are
EGel and note that, by Lemma 16.10,
P is
Applying Step 1, we deduce that there is an irreducible such that P is a vertex of
in B'
by (4) JCFGlV
we
i s FG-projective.
Here we complete the proof by establishing (31 f o r an arbitrary
Step 3. Let e
1
Thus
VFG
V.
Now J(EG)V
=
0
and so,
is completely reducible and so S
VFG
= .@ Mi
z=1 for some irreducible FG-modules identity on
V,
Mi
all
16.11 ensures that V
Mi.
since multiplication by
are in B .
By Step 2,
V
i s
e
induces the
FG-projective, so Lemma
is a direct summand of S
EG 8 V
=
FG V
But ME
is irreducible, so V
{ M ,,..,,M 8 1 .
FG 8 M
and thus
@ LEG @ Mi) i=l FG
is a direct summand of EG 8 M
for some
FG Denote by
Q a vertex of M.
Then M
i s a direct summand of
V is a direct summand of
FQ EG 8 (FG 0 M ) FG FQ Applying Lemma 16.11, we infer that projective and thus
is FQ-projective.
P is conjugate to a subgroup of Q.
ducible FG-module in B .
.
jugate to a subgroup of wished to show.
V
EG 8 M FQ
Consequently M P.
A fortiori, V Now M
is P-projective and so
This proves that P
and
is EQ-
is an irre-
Q is con-
Q are conjugate, as we
As a final preliminary result, we prove
16.15. LEMMA.
Let N
be a normal subgroup of
G and let V be an FGmodule.
Then the following properties hold: (i) J(FN1V (ii)
If
V
is an FG-module is N-projective, then
V/JCFGIt: is N-projective if and only if
200
CHAPTER IIL
V
(iii) If V/SCFG) V
is projective and indecomposable, then N
contains a vertex of
if and only if
Proof.
(i) For a given g E G ,
and hence of JCFN).
g
conjugation by
is an automorphism of FW
Bearing in mind that
we infer that FGJCFN) = J(FN)FG.
Hence
as required. (ii) Put M J(FN1V
=
5 3(FG)V
Because 'Cv/J(F#)VIN
f :M
-+
-
J(FGlV/SIFN1V
--t
V/J(FN)V
--+
M
-
The inclusion
0
is completely reducible, the above sequence splits as an
But M
V / J ( F N )V
is N-projective.
induces an exact sequence
0
FN-sequence.
and assume that M
V/J(FG)V
is N-projective, so there exists an FG-homomorphism
such that
Bearing in mind that M
Hence J ( F G ) V / J ( F N ) V
=
0
and f ( M )
are completely reducible, we derive
and J V G ) V = J ( F N ) V.
Conversely, suppose that J V G ) V
=
J V N )V.
Since A = J ( F N )
condition of Lemma 16.13, it follows that M = V/JVG)V= V//s(FN)V
satisfies the is N-projective
THE RADICAL OF BLOCKS
(iiil
281
Direct consequence of ( i t ] .
We are at last in a position to achieve our main objective, which is to prove the following result. 16.16. THEOREM CKnorr (1976)).
be a normal subgroup of G
with defect group D.
be a block of FG
B = B(e)
Let N
and let
Then the following are
equivalent: (i) D
5N
(ii) [J(FG)lnV = [J(FNllnV for all indecomposable FG-modules V
in B
and all
positive integers n (iii) J ( B ) = S(FN)B for all projective indecomposable FG-modules V
JCFGIV = JcFN1V
(ivl
Proof.
(i) * (iil:
By Theorem 16.14, V
Assume that
V
is an indecomposable FG-module in B .
is D-projective and so
v
is N-projective.
conclusion now follows from Lemma 16.5Cii) by induction on (ii) * Ciiil: Write
e
=
+
e
... +
The desired
12.
n as a sum of primitive idempotents of FG.
e
is an indecomposable module in B
Then each FGei
in B .
and so
Bearing in mind that S(FGle
=
J(FGlel +
... + J ( F C l e n = J(FGIFGel + ... + J(FG1FGe,?
we derive
n (iii)
* (iv): Write B
=
@
Ui , where the U. are projective indecomposable
i=1
modules in
B.
Then
282
CHAPTER I11
Since
5S ( F G I U i
J(FN)Ui
i,
for a l l
w e have
J (FG) Ui
Hence (iv)
B
*
=
J (FN)Ui
J(FG)U = J(FN)U f o r a l l p r o j e c t i v e indecomposable modules ( i ) : Owing t o Theorem 7.6.7,
D is
such t h a t
U i n B.
composable FGmodule
'I
B
in
V
t h e r e e x i s t s an i r r e d u c i b l e FG-module
V.
a, v e r t e x of
in
U/J(FG)U f o r some p r o j e c t i v e inde-
V
Now
i
for a l l
The d e s i r e d conclusion i s now a consequence of
.
Lemma 16.15 (iii) 16.17. COROLLARY.
B
J(FD1
=
I(D),
16.18. COROLLARY. i s a block of
Proof.
with normal d e f e c t group
Define
FG
Put
0 (G1 PIP'
1= 0
(G)
and
D
P
=
N.
=
0 G I / O (G) PIP'
by
with d e f e c t group
N
D,
we have
D.
Then
J ( B ) = J(FD)B.
'
t h e r e s u l t follows.
contained i n
0 (G).
(G/Op(G))
0
=
If
B
0 (GI, then PIP'
i s a normal Sylow p-sub-
J(FN) = F N * I ( P ) .
Hence, by Proposition 1.1,
-
P'
P
Then
P
PIP'
group of
FG
be a block of
By Theorem 16.16, applied f o r t h e c a s e
Proof. Since
Let
Since
D
5N,
the
r e s u l t follows by v i r t u e of Theorem 16.16. Recall t h a t
G
is c a l l e d p-solvable i f each of i t s composition f a c t o r s i s
e i t h e r a p-group o r a P'-group.
G
Thus
i s s o l v a b l e i n t h e ordinary sense i f
and only if i t i s p-solvable f o r a l l primes
p.
G
Clearly
is p-solvable i f and
only i f it has a s e r i e s of normal subgroups
1 = V
C V
0 -
f o r which each f a c t o r group
vi-l/Vi
C . . . C V
-
1 -
n
= G
(4)
i s e i t h e r a p-group o r a p'-group.
For
each such group we can t h e r e f o r e d e f i n e t h e upper p-series
l = P Clv 0 - 0
C P 1
C P C . ..CP 2
C N
R-
9.
= G
(5)
283
THE RADICAL OF BLOCKS
i n d u c t i v e l y by t h e r u l e t h a t
(G/P,)
Nk/Pk = 0
Pk+l/Nk
and
=
P'
0 (G/Nk)
P
Thus, i n p a r t i c u l a r ,
No
(G)
= 0
, PI
=
0
P'
R
The number
and
N
=
and i s denoted by
(G)
0
P '*PIP'
which i s t h e l e a s t i n t e g e r such t h a t
G
l e n g t h of
(G)
P'rP
!L ( G I .
NR = G i s c a l l e d t h e p-
Observe t h a t
P
!L (G) may a l s o be defined
P
a s t h e s m a l l e s t number of p - f a c t o r s occuring i n any series such a s (4), t h e minimum being a t t a i n e d f o r t h e upper p-series ( 5 ) .
Thus
G
i s of p-length
1
i f and only i f
G/O
(GI
i s a p'-group
P'tP With t h i s information a t our d i s p o s a l , we now prove 16.19. COROLLARY.
PE
Syl ( G I .
G
Let
be a p-solvable group of p-length
Then, f o r any block
P
J(B)
B =
1 and l e t
FG,
of
G))B
S(F0
P'rP Proof.
N
Put
p-subgroups of
G.
=
0 GI. P'nP
Since
G
i s of p-length
1, N
contains a l l
Now apply Theorem 16.16.
To p r e s e n t our f i n a l a p p l i c a t i o n of Theorem 16.16, we quote t h e following standard group-theoretic f a c t . 16.20. PROPOSITION. Then
G
i s of p-length
Proof.
16.21.
be a p-solvable group with a b e l i a n Sylow p-subgroups.
.
1.
See Gorenstein (1968,p.228).
COROLLARY.
and l e t
G
Let
Let
P E Syl (GI. P
G
be a p-solvable group with a b e l i a n Sylow p-subgroups,
Then, f o r any block
J(B) = J(F0
B
of
FG,
(GI ) B
P'rP Proof.
Apply Proposition 16.20 and Corollary 16.19.
.
CHAPTER 111
284
17. GROUP ALGEBRAS WITH RADICALS EXPRESSIBLE AS PRrNClPAL IDEaLS Let G
be a field of characteristic p > 0 .
be a finite group and let F
The
problem that motivates this section is to discover conditions under which J ( F G ) = ccFG
FGx
=
for some z E F G .
The main result, due to Morita (1951),
solves this problem for the case where F
is algebraically closed.
We have
chosen, for the sake of clarity and generality, to deal with the problem within the framework of a general theory, namely the theory of uniserial algebras, although a shorter proof can be obtained by means of ad hoc arguments. Throughout this section, A arbitrary field F.
denotes a finite-dimensional algebra over an
Unless explicitely stated otherwise, all A-modules are
assumed to be left and finitely generated. Let
V denote an A-module.
Then the dual module
V*
=
Hom(V,F) is a right
F A-module with the action of A
Tf
on
V*
given by
V is a right A-module, then a similar definition gives a left A-module
The following properties of left Lor right) A-modules V
and
W
P.
are easily
verified : (a)
V is indecomposable if and only if V*
(b)
V
(cl
(V**)*
(dl
(V @ U *
V*
is projective if and only if 2
is indecomposable
is injective
v 1
V* @ WY
Let V be an A-module.
of submodules of
V
Then the descending chain
is called the (lower1 Loewy series of
nilpotent, there is an integer k ,
V.
called the Loewy length of
Note that if J ( A ) ~ V= J ( A ) { + ~ V for
i < k , then
Because J ( A )
V
such that
is
GROUP ALGEBRAS WITH RADICALS EXPRESSIBLE AS PRINCIPAL IDEALS
a contradiction.
k
The conclusion i s t h a t , i f
285
V,
i s t h e Loewy l e n g t h of
then
of
V.
i s s a i d t o be uniseriaz i f t h e above chain i s a composition s e r i e s
V
The module
V
Thus
i s u n i s e r i a l i f and only i f i t has a unique composition s e r i e s .
Because a u n i s e r i a l module has a unique i r r e d u c i b l e submodule, it must be
indecomposabte.
We s h a l l r e f e r t o t h e a l g e b r a
indecomposable A-module i s u n i s e r i a l .
i s a u n i s e r i a l r i g h t module.
A
u s being uniseriaz i f every
Note t h a t t h e dual of a u n i s e r i a l module
Hence, i f
A
i s u n i s e r i a l , then every indecom-
posable r i g h t A-module i s u n i s e r i a l . 17.1. LEMMA.
Let
V be a ( l e f t ] module over an a r b i t r a r y r i n g R.
Then the
following c o n d i t i o n s a r e equivalent.
(il
V
is i n j e c t i v e
(ii) Every homomorphism
from any module containing
(iiil
If
T
V
W--+
o f R-modules can be extended t o a homomorphism
W to
is a l e f t i d e a l of
V.
R
and
4
E Homu,V1,
then
4 can be extended
R Hom(R,V). R (i) * ( i i l : Assume t h a t
t o an element o f Proof.
homomorphism of R-modules.
If
X
V
@
I s i n j e c t i v e and t h a t
i s an R-module containing
W
:
v-
and
V c1 :
i s t h e n a t u r a l i n j e c t i o n , then by d e f i n i t i o n t h e r e e x i s t s an R-homomorphism
$*
:
X+
V which renders commutative t h e following diagram:
\
is a
W+
X
286
CHAPTER I11
Hence $*
(ii) * (iii): (iii)
4.
extends
Obvious
* (i) : Assume that
v
satisfies (iiil and let X,Y,a,f3
be given as in the
following diagram
B
Put CC(X1
yo
:
Xo
=
xo-
Yo
V by
=
6a-l.
Now let 2
such that U
is a submodule of
Cu1 ,Y1 1 , (U2,Y
1
and y
extends yl. 2
=
an R-homomorphism @
{r E RlPy :
T-+
$
extends to an R-homomorphism
Y*
:
U+@
by
y E U,
Y* (tr+pY)
and because y
17.2. LEMMA.
Let
Set
M
E
(U,yl.
y
extends y
0'
given by
@* : R-+ V . + @*(r1
We show that
V be an A-module. V
u
=
If
Y.
Then 1 is a left ideal of R
Uj.
was arbitrary we have
= HomU,Vl
and
$(r1
=
and
y ( ~ y ) . By hypothesis,
With this map we can now define
.
This is easily seen to be well
Hence by the maximality of
injective A-module that contains Proof.
k'
= Y (tr1
CU,Y1 G (U+Ry,Y*1.
defined and
v,
(U1, y l l G ( U 2 , Y 2 ) if and only if U 1 c -U
contains a maximal element, say,
r
U-+
:
(u,Y)
(~o,yol E 2, it follows easily from Zorn's lemma
Since
Let y E y and let we have
be the set of all ordered pairs
Y,u 3 xo, y
are in 2, we write
2
that
a is injective we can define
and observe that because
U = Y.
(U,Y) we have
so the lemma i s true.
Then there exists a (finitely generated) as a submodule.
and note that
M is an A-module via
F
We claim that $ : I--t
M
is injective.
M be an A-homomorphism.
Indeed, let I Define
:
be a left ideal of A I+
V by
and let
207
GROUP ALGEBRAS WITH RADICALS EXPRESSIBLE AS PRINCIPAL IDEALS
p
Then
is c l e a r l y a n F-homomorphism and hence we may extend
h
morphlsm
0
Then
:
V.
A-
Now d e f i n e
M by
$
and so
i s an A-homomorphism extending
V
Because t h e map
17.1.
0 : A-
--+
v
M given by
M
If
A
is a Frobenius a l g e b r a , then
f,(a) = av,a E A,v E V ,
with
i s an i n j e c t i v e A-homomorphism, t h e r e s u l t follows. 17.3. LEMMA.
is i n j e c t i v e , by Lemma
.
fv
A
t o an F-homo-
?A
i s an i n j e c t i v e l e f t A-
module. Proof.
Let
4
1 be a l e f t i d e a l of
and l e t
f
E Hom(1,A).
By Lemma 17.1,
A it s u f f i c e s to show t h a t
thesis, there exists
a E A,
define
clearly
ha
ha:
Q
f
E A*
r-A
can be extended t o an element of such t h a t
1a ( x )
by
(A,$)
= xa.
M
=
To t h i s end, note t h a t t h e mapping with k e r n e l
1’
{hala
dim(Hom(I,AI 1 F
Then each
Given
Horn(1,A)
and
of
End(A) A
by p u t t i n g
=
x;(Z)
ax2
E A)
1.
If dimV = 1, then the assertion is obvious.
F If
V
F
is irreducible, then V
is obviously uniserial.
V has a proper irreducible submodule uniserial.
Denote by
W
By induction hypothesis, V'
8'.
W # 0.
respect to the property that
PI n X
must be
V with dimW as large as
a uniserial submodule of
Then we must have
possible.
Thus we may assume that
F Let X be a submodule of V, maximal with =
0,
Assume that X /X and X /X are irre1
ducible submodules of
XI n X2 n
w
# 0
since W
proving that Soc(V/X)
Then X n
V/X.
is uniserial.
W#
0
W # 0
and X
Hence X 1
and hence
n X # X and so X 2
is irreducible. By Lemma 17.5(ii),
V/X
of an indecomposable injective A-module, and so is uniserial.
exists a homomorphism g : p-+
V
X , 2
is a submodule
Invoking Lemma
17.7, we deduce that there is a surjective homomorphism f : P-+ V/X suitable indecomposable projective A-module P.
= 1
for a
Since P is projective, there
which renders commutative the following diagram:
291
GROUP ALGEBRAS WITH RADICAL EXPRESSIBLE AS PRINCIPAL IDEALS
Because P
is uniserial, so is g(P).
isomorphic to a submodule of
V/X.
w
Now
n
x=
0 implies that kr is
Therefore, by the maximality of
dimW ,
F
dimg(P)
F
dimW F
dim(lf/XI F
On the other hand, the commutativity o f the diagram ensures that dimg CP) 2 dim tV/XI F F
W.
Y = gCP)
+X
and g ( P ) n X = 0.
Hence g ( P )
V/X
V
and the desired assertion follows by induction hypothesis.
=
g(P) @ X
17.9. COROLLARY.
Let
It follows that
A be a Frobenius algebra.
only if for each primitive idempotent e
of
Then
Thus
A is uniserial if and
A , the A-module
Ae
is uniserial.
Proof. Apply Theorem 17.8 and Lemma 17.6. 17.10. LEMMA.
in
A.
Let I
be an ideal of A
such that I = Ax = yA
for some x,y
Then I = 4 = Ay.
Proof. Given an A-module The map A
d I, a
I-+
ax
Hence c ( I ) = c(A/k(dAl I .
M, let c W I
denote the composition length of M.
is a surjective A-homomorphism with kernel k(x) = k ( d ) . However, 4
c - I and k ( d ) 2 %(I]. Thus
CHAPTER I11
292
By combining these two i n e q u a l i t i e s , we deduce t h a t
I = Ay
I
and, by a similar argument,
C(T) =
c(Ay).
Consequently,
So t h e lemma i s v e r i f i e d .
= xA.
We have now a t our d i s p o s a l a l l t h e information necessary t o prove t h e following important r e s u l t . 17.11. THEOREM. (Morita (1951)., Azumaya and Nakayama (1948)1.
F.
algebra over an a r b i t r a r y f i e l d (i) There e x i s t
(iil
x,g
x EA
There e x i s t s
(iii) A
A
in
be an
Then t h e following conditions a r e equivalent:
J(A1 = xA
such t h a t
Ay
J ( A ) = xA =
such t h a t
A
i s u n i s e r i a l and each block of
Proof.
A
Let
Clearly ( i i ) implies (il.
=
Ax.
i s quasi-primary.
By Lemma 17.10,
( i ) implies (ii)and
hence we need only v e r i f y t h a t l i i ) is equivalent t o ( i i i ) . For t h e sake of c l a r i t y , we d i v i d e t h e r e s t of t h e proof i n t o t h r e e s t e p s .
Here we prove t h a t (iil impZies t h a t
Step 1.
J ( A ) = XA = AX
Assume t h a t idempotent of
A.
Let
of
a 9 J(A)l’+le.
a. 4 J ( A ) e ,
Aa
=
Ae,
such t h a t
a
SL4)e
since
=
J(Alpe Because
x’a0.
submodules of of
eA.
I = J(A)”e.
Ae.
Hence
S t e p 2.
A
and
e
be any p r i m i t i v e
JIA)’e = x’Ae,
Then
Ae.
17 - S(Al’+’e. Aao
5 Ae
=
Ax’a
=
Let
This shows t h a t
Similarly
Ae.
Hence t h e r e
a
there exists and
Aao Ae.
Then
be an element
a E Ae,
J ( A ) e , so t h a t Hence
S(A)’ao = J(A)l’e A e , J ( A ) e , J U ) ’ e , . . . a r e t h e only
eA,eJ(A) , e J ( A ) ’ , . .
.
exhaust a l l r i g h t submodules
i s u n i s e r i a l , by appealing t o Theorem 17.8.
We now prove t h a t liil implies (iiil.
By Step 1, we need only show t h a t each block of end, w r i t e
and l e t
A,
i s a unique maximal submodule of
I 2. Aa proving t h a t
in
i s a unique maximal submodule of
FI 2 1 such t h a t T
I with
x
I’ be any proper nonzero submodule of
I c J ( A l e s i n c e J(A1e exists
f o r some
i s uniserial.
A
A
i s quasi-primary.
To t h i s
293
GROUP ALGEBRAS WITH RADICALS EXPRESSIBLE AS PRINCIPAL IDEALS
where t h e
i
e..
23
a r e mutually orthogonal p r i m i t f v e idempotents,
E {l, ...,f ( k ) } and
JUI’(= Ax’ =x”A) i s
Aeki ir Ae
Aeki
Aek,
if
k # t.
for
Ax’eki:
Ae . .x’ 13
i E [1,2,.
for a l l
Aek,
Then i t i s e a s i l y seen t h a t
e x p r e s s i b l e as a d i r e c t sum of
and s i m i l a r l y a s a d i r e c t sum of
Because
ti
Aeki
. ., f ( k ) 1,
we have
and
Ae tix’ where
Ax’e
ki
Ae .xu tJ
and
with
Ae tlx’
Ableki
= 0
and
(1
Ae x’ tj
=
j
Q
f(t))
0 a r e excluded.
Next, we prove t h a t
Ax’eki
Ax’etj
Ae x’ 2( Aekza’ tj Ax’eki where
Ax’eki
Ae x’ ti
k # t
if
t # k
a r e indecomposable l e f t i d e a l s
Ae .xu with Ax’eki = 0 tJ Then, by Ax eki zz Ax’etj.
and
Suppose t h a t
and
if
and
’
J(,4)’eki/J(A)’+leki
Ae x’ tj
=
0 a r e excluded.
S t e p 1, t h e r e e x i s t s
esm
such t h a t
Aesm/J(A)esm
and
J(A)’etj/J(A)’+’etj Choose
a E esmJ(A)’eki,b E esmJ(A)’e
tj
1
with
Aesm/J(A)esm a 9 J(A)’+l,b 9 J(A)’+l*
have
e JW’= sm
a~ = b~
Then w e
294
CHAPTER I11
and therefore e
J(A1’
is a homomorphic image of both
srn
e A kz
e A tJ
and
But
then
ekiA/eki J U ) and hence
k = t,
d
e t jA/e t j J ( A )
proving ( 5 ) .
Property (6) is a direct consequence of the fact that Ae
To prove (71, note that Axveki
morphic image of A e t j . of some Ae and Ae
ti
sm
x”
and Ae x’
tJ
x” is a homotj is a homomorphic image
”
is a homomorphic image of A e t j .
Thus both A X eki
are uniserial and therefore indecomposable, proving ( 7 ) .
Invoking ( 3 )
-
( 7 ) and the Krull-Schmidt theorem, we deduce that there is a
one-to-one correspondence
$
between the subsets of
{1,2,.
. .,n}
such that
and
and Ae xu which reduce to zero. tJ From (8) it follows further that
where we exclude those Ax’ekz.
if h u e k l # 0 ,
then JL41’ekl/J(d)”’1ekl
Hence the number f ( k ) A.
is the same f o r any Aeki
This proves that each block of A
Step 3.
A e @ ( k ,)l / J ( A ) e $ ( k ),1
( 10)
contained in a fixed block o f
is quasi-primary.
CompZetion of the p r o o f .
By Step 2, it suffices to show that Ciiil implies (ii1.
Then the proof of (5) shows that k # t
implies
To this end, we put
$(k) # $(t).
Since Aekl
and
are contained in the same block, we have f ( k ) = f ( $ ( k ) ) , since each Ae $ ( k l ,1 block of A is quasi-primary. For an element C k i such that
we have
GROUP ALGEBRAS WITH RADICALS EXPRESSIBLE AS PRINCIPAL IDEALS
and ck.;A = e
Acki = J(Aleki
n Setting x
f(kl
Z
C cki k=1 i = 1
=
295
(11)
$ ( k ) ,i J ( A 1
, we obtain a direct decomposition n Ax =
f(k) Z Acki
C
k=1 i=1 Applying (ll), ekiA = e
$ ( k ) ,%
we infer that J ( A ) = A X .
.JtA)
and J ( A )
=
xA.
Similarly xA
Hence J ( A )
=
Ax
is a direct sum of
X A and the result
=
follows.
For the r e s t of t h i s section,
F
denotes a fieZd characteristic p > 0 . FG
next aim is to discover necessary and sufficient conditions under which finite representation type.
sum of elements in X
Given a subset X
and denote by
FX'
of
we write 'X
G,
the F-linear span of .'X
our is of
for the The next
three lemmas and Theorem 17.15 (below) will not be required in their full generality. Let G be a p-group.
17.12. LEMMA.
(il
If
I is a proper left ideal of F G ,
then F G / I
is an indecomposable FG-
module, (iil
FG has a unique minimal left ideal, namely FG'
(iiil Each nonzero submodule of FG Proof.
(il
is indecomposable.
Owing to Corollary 1.2, J V G )
=
Hence J(FG)/I is a unique maximal submodule of
I(G)
and so F G / J V G )
FG/I
and thus F G / I
F. is inde-
composable. (iil
It is obvious that FG'
is a minimal left ideal of FG.
is an arbitrary minimal left ideal of FG. afforded by
Since the representation of
X is irreducible the isomorphism FG/J(FGI
acts trivially on X for all g E G.
and d i d = 1.
F + But then x = XG
Hence X
=
Fx
for some h # 0
P
F
in F.
(iii) Direct consequence of (iil. Suppose that G = < g >
is cyclic o f order pn.
X
G
ensures that G
for some x E FG
asserted.
17.13. LEMMA.
Assume that
and gz=z
Thus X =' % 2
as
296
Iil
CHAPTER I11
Each indecomposable FG-module is a homomorphic image of FG
i E {l,2,..,,pn},
(ii) For any
has exactly one submodule of codimension i,
FG
namely J(FG)'
n
(iii) For any i E {1,2 ,...,p },
put
Vi
= FG/cJ(FG)~.
Then
(a) V .
is an indecomposable FG-module of dimension i, and
F-basis
Iv 1 ,...,v 2. } such that
(b) Each indecomposable FG-module is isomorphic to
n Proof.
ti1
We first note that FG
principal ideal domain, each FIX] cyclic FIX]-submodules. is cyclic.
Hence, if
4
F[Xl/@
Vi possesses an
,...,p n } .
V;
for some i ~ i 1 , 2
- 1).
Because FIX1
is a
(and hence FGI-module is the direct sum of
V
is an indecomposable FG-module, then
It follows that there is a surjective homomorphism
FG-+
V
1/
PG-
of
.
modules, proving (i)
correspond to the ideals 1 of FIX] for which
(ti) The submodules of FG
n
13 urp
-11
n Since FIX] is a principal ideal domain, I = cfl for some f dividing X p n n But f - 1 = (X-11' , so T = I I X - 1)' for some i, 0 Q i 4 p n , and the image
I* of T in FG is .T* = F G l g - 11 i
Because JVG) = T ( G )
and
Hence the submodules of FG
and
G is generated by 9 ,
are the
we
also have
-1
297
GROUP ALGEBRAS WITH RADICALS EXPRESSIBLE AS PRINCIPAL IDEALS
as asserted. each V . z
(iii) Owing to Leunna 17.12(iI,
..,V
we obtain an F-basis
proving (a).
{Ul,.
Assume that
By ( i l ,
is an indecomposable FG-module.
V
FG/I
i for some i and hence
Let G = < a > x ,
17.14. LEMMA.
where < a >
Let
and < b > are cyclic groups of there exists an indecomposable FG-
Then, for any positive integer n,
module of dimension Proof.
Vi such that
.
(b).
order p .
of
Setting
By lii) , .T = FGLg - 1)
for some FG-submodule I .
proving
V
.I
is indecomposable.
2n+l.
V be a vector space over F of dimension & + l
and let
... ,xn y ,... ,Y, 1
‘xo ,t , be a basis of
V.
I
~
It is inmediate to verify that V becomes an FG-module by
setting
Let X and
Y denote, respectively, the F-linear span of { x ~ , ...,Xn 1 and
{Y~,.-.,Y~], and let
?T
:V-
Y be the natural projection.
left multiplications by (a- 1) and
Owing to U2),
db - 1) provide injections Y-+ X and map
to 0 Assume by way of contradiction that
V
=
V @ V 1
2
for some nonzero FG-sub-
x
CHAPTER I11
298
modules
v1
and
v2
of
v,
Put r
(a-1)V C V 1 -
=
and 1
dim.rr(V1)
SO
F
(a-1)V
that 1 g r
< n.
Then
= (a-l)7T(V1)
and therefore dim(kz-1)V
= P
F Bearing in mind that
(a- 1)V1, we derive
(b - 1)V1
(0 Q P Q n)
dimV 2 2 r + 1
F’ A
similar inequality is also true for
.
Setting
3 =
dimv(V 1 , F 2
we have
r + s > n and dlmV2 2 2s + 1
F Hence dimV 2 2 ( r + s l
+
2
? 2n
+2
F
.
contrary to the assumption that dimV = lemma.
2y1+
This completes the proof of the
1.
F
It is now an easy matter to prove the following general result. 17.15. THEOREM, (D G Higman (1954), Kasch, Kneser and Kupisch (1957)). be an arbitrary field of characteristic p of
3
0
Let F
and let P be a Sylow p-subgroup
G.
(i) If
P is not cyclic, then there exist indecomposable FG-modules of arbitrary
large F-dimension (ii) If P
is cyclic, then there exist at most
posable FC-modules. Proof.
IGl
nonisomorphic indecom-
Any indecomposable FG-module is a homomorphic image of
(i) Assume that
P is not cyclic.
abelian factor group of order p2.
(flP.
P has an elementary
Hence, by Lemma 17.14, for each n 3 1 there
exists an indecomposable FP-module Vn rnorphic to a direct summand of
Then
FG.
with dimV = 2 n + l . But Vn is isoF n Hence, by the Krull-Schmidt theorem,
Vn
GROUP ALGEBRAS WITH RADICALS EXPRESSIBLE AS PRINCIPAL. IDEALS
is isomorphic to a direct summand of
Ifil.
Y of
299
Wp for some indecomposable direct summand
It follows that dimW 2 dimVn = 2n + 1,
F
F
as required. (ii) Since P
is cyclic, it follows from Lemma 17.13 that there are
,
composable FP-modules V 1 , V2,...
where
dimV.
=
i
\PI inde-
and every indecomposable
F" FP-module is isomorphic to some module.
(V,)
G
.
Write V
P
= @
83
is an indecomposable FG-
is isomorphic to a direct summand of
F
with j
V
X i for some indecomposable FP-modules Xi.
Consequently, V
dimV 2 didi.
F
Assume that
V is P-projective, so V is isomorphic to a direct summand of
Then
mull-Schmidt theorem, V and
Vi.
Then, by the
4
for some i
is isomorphic to a direct summand of some
fl
< dimV.
Since d i m e 3
F
composable direct sununands o f
fi3
CG:P)J', the number of nonisomorphic inde-
=
of dimension at least j
(G:PI.
is at most
Hence the number of nonisomorphic indecomposable FG-modules is at most By Lemma 17.3(i), morphic image of Thus V
Vi is a homomorphic image of FP.
(FPIG
FG.
But
V
Hence
vGi
\GI.
is a homo-
is isomorphic to a direct summand of
is a homomorphic image of F G ,
83
as asserted.
We have now come to the demonstration for which this section has been developed. 17.16. THEOREM (Morita (195111. characteristic p group of
G.
3
Let F be an algebraically closed field of
0, let G be a finite group and let P be a Sylow p-sub-
Then the following conditions are equivalent:
(i) There exist x,y E FG such that JCFG) = zFG = FGy (ii) There exists x E FG (iii) G Civ) (v)
such that JlFG) = zFG = FGz
is p-solvable and P
is cyclic
PO , ( G ) a G and P is cyclic
P
PO ,(G) d G , P is cyclic and G/PO ,(G) P P Proof.
is cyclic of order dividing p
(i) * (ii): Direct consequence of Lemma 17.10.
(ii) * (iii): By Theorem 17.11, FG
is uniserial and each block of FG
is
- 1.
CHAPTER I11
300
quasi-primary.
Hence, by Lemma 17.7 and Theorem 17.15, P
Proposition 14.5, PO f ( G ) d G.
is cyclic and, by
Since the latter implies that
P
G
is p-solvable
(iii) is established. tiii) * (iv): By Proposition 16.20, G
PO , ( G I P (iv) * (v): Set C(P) =
5
=
since CLPI
G
G
subgroup of
Aut (PI.
G/O
P
(GI
a cyclic group of order
12.1.
i;
and
P P and
But
p-1.
p
n-1
o
(GI a G
=
PO
=
1.
P
Hence, if x E FN
P
,(GI.
:/F
Hence
z/p
=
G/PO ,CG).
pn
and p > 2,
(p-11.
Therefore G/PO , ( G I
=
P
we have
G
=
is such that J V N )
B be an arbitrary block of FG. Since P
G/pO
P
(G)
is
then Aut(P)
is
is a cyclic group of
PO f ( G ) , by virtue of Lemma
=
FNx
=
P
= 2.
xFN, then
=
FGtFNx) = (xFNIFG
=
FGx = xFG in which case G
is p-nilpotent.
Let
It suffices to verify that J(BI = xB = Bx
for
is cyclic, it follows from Corollary 10.10 that
M (FD) for some cyclic p-subgroup D of G.
Since J(B)
since
where g
and
and observe that, by Proposition 1.8(ii),
Thus we may harmlessly assume that G = N
some x E B.
Thus
P
IP(
45
is isomorphic to a
If
In case p = 2
PO ,(GI P
Then
Aut(P).
J(FG1
B
,(G)/O
Thus the required assertion is also true for p
(v) * (ii): Put N
1 and so
P'#P
a 5 and 0 ,(:I
isomorphic to a subgroup of
order dividing
=
is of p-length
is a generator of D, the result follows.
9
Mn(J(FD))
and
301
4 Group algebras of p-groups over fields of characteristic p In this chapter we confine our attention to the case where G a field of characteristic p .
and F
After examining dimension subgroups, we
provide Jennings' description of bases for S ( F G I n tency index
t(G1
of S ( F G ) .
and a formula for the nilpo-
As an application, we determine the right and left
annihilators of powers of I l G )
and exhibit a remarkable symmetry of the numbers for some individual p-groups P,
= did(Gln/l(G)n+l. We then compute t ( P ) n F such as metacyclic p-groups, p-groups of order
c
p4, etc.
I n the final section, all p-
second highest, lowest and second lowest values.
t(P)
7
are characterized.
p
I. DIMENSION SUBGROUPS IN CHARACTERISTIC Throughout this section, F G
and
for the subgroup of
G
3
0 ?&ID RELATED RESULTS.
and
G2 of
G,
The lower c e n t r a l s e r i e s
{y,(G)}
-1 -1
= z
of
G
y
with
xy
LC
E G
is central in
nilpotent if Yc+l(G) est such integer G.
If
= 1
G/Y,~(G).
for some integer
c with Y,+l(G)
= 1
1
,y
E G2
G
and
is defined inductively by
It is clear that these groups are, in fact, characteristic subgroups of Y,(G)/Y,~(G)
IG1,G21
we write
generated by all commutators
IXPYI
that
p > 0
denotes an arbitrary field of characteristic
Given subgroups G I
a finite group.
The results obtained
P in case t ( P ) attains its highest,
are applied to investigate the structure of
groups P with
is a finite p-group
Furthermore, c 2 0.
G
is called
When this occurs, the small-
is, by definition, the nilpotency c h s s of
n 2 1 is an integer, we put Gn = < g n l g E G>
CHAPTER 4
302
It is clear that all such subgroups Gn
are characteristic in G.
The Erauer-Jennings-Zassenhaus series for G
M
1 ,P
(GI
is defined inductively by
= G
and for n 2 2
M
nI
(G) = lMn-l
(GI , G I M i , p ( G ) P IP
P
is the smallest integer satisfying i p 2 n ,
where
One immediately verifies,
by induction, that this is a decreasing sequence of characteristic subgroups of
G. The Lazard series for G
is defined by
The following observations are easy consequences of the definition: (a) L
1I
(G)
G
=
P
(b) L q l p ( G )
is a product of finitely many of the characteristic subgroups
Y i (GI p3
is a decreasing sequence of characteristic subgroups of
(c) { L n , p ( G l }
G.
(GI contains the factor Y1 (G)" = G and so L (G) = G. The #P 1 tP reason for (b) is that, since L I G ) >_ Y n ( G I P 0 = Y n [ G ) , all additional factors n,P with i n are redundant. A l s o , for i n only the minimal j with ip' n Indeed,
L1
is really required,
Finally, (c) is a consequence o f the fact that increasing n
decreases the number of defining factors for L The dimension subgroups
Let G = G G.
Then
1
>_
G
3
2 -
Dn(FG)
of
... 3 Gn -3 ...
G
(GI.
are defined by
be a decreasing sequence of normal subgroups of
is said to be an N -sequence if the following two conditions hold:
{Gi}
P
IGitGjl
g E G~ Of course, if
n,P
G
Gi+j
implies g p E G
iP
is a p-group, then the above implies that Gi/GPi
(and hence
DIMENSION SUBGROUPS
G
3 Hn+l
= 1
1 choose
gi E H.-Hi+l.
Then t h e
elements
x (al,a2,... ,an) =
n a n (gi-u i i=l
with
0 6 ai < p
h(0,0,
...,01 = 1
Proof.
form an F-basis f o r
FG.
Moreover, t h e s e elements with
d e l e t e d form an F-basis f o r
I(G).
Since t h e number of t h e given elements
h(al,...,an
i s precisely
CHAPTER 4
304
G,
the order of
it suffices to prove that they span FG. \GI.
being trivial, we argue by induction on
The case G = 1
IH21 < I G l ,
Since
it follows
from the induction hypothesis that the elements
span FH
.
Now
is cyclic of order p
G/H2
,...,
and is generated by the image of
is a transversal for H 2
C I G = 1 - d - d+l
2 -
For each g # 1 in G ,
G.
for which g E Gm, and put
to be the largest m
W(g1,
,...,a
)
define the height of g,
is as in Lemma 1.2, we define the weight
W(A)
V ( 1 ) = m.
of
X
If
by
n k Let Ti be the F-linear span of all products of the f o n ( ~ ~ - 1for ) some k k j=1 with 5 E G and c V(x.1 2 Clearly, Ii is an ideal of FG because it is
n
i
j=1
G
1 -
=
,...,a
P
-sequence
- d -3 Gd + l = l
I(G)
determined by t h i s sequence.
i s as i n Lemma 1 . 2 ,
X(a
N
3 . . . > G
2 -
be t h e f i l t r a t i o n of (a
If
305
then t h e elements
with
w ( A ) 2 t form an F-basis f o r
with
Note f i r s t t h a t , by d e f i n i t i o n of
W(x)
It, i f
and
W ( x ) =
It
=
W(x1
t
t,
W e a r e t h e r e f o r e l e f t t o v e r i f y t h a t any product
E Tt.
(x.E G) 3
k
c
with
i
V ( z .)
j=1
can be w r i t t e n a s an F - l i n e a r combination of
3 plus a term i n
We argue by induction on
i E 11,...,k l
To begin w i t h , we f i x
a,B E FG.
Put
refinement of
V(x.1
z
{Gs},
f o r some i n t e g e r
r.
=
m
so t h a t
t
k.
and w r i t e
g E G,.
'S with W ( x ) =
u = C&(z.-l)@ f o r some
Since t h e sequence
{Hsj
is a
w e have
Invoking Lemma 1 . 2 , w e i n f e r t h a t
x.-1
z
i s an F-linear
combination of terms of t h e form
with t h e
as
n o t a l l zero.
However, each
gs
above s a t i s f i e s
VCg
)
2 m
= v(x .)
and so
f o r some
bs E F,y E T,+l.
Hence
and t h e r e f o r e it s u f f i c e s t o v e r i f y t h a t each product combination of t h e each
zi-1
height.
If
A's
modulo
It+l.
Moreover, by applying t h i s argument t o
i n t u r n , we may assume t h a t each
k = 1, then
u = g
j
-1
a ( g -1)8 i s an F-linear
xi i s
f o r some
j
some
with
gi,
with t h e same
V ( g . ) 3 t. 3
Hence
CHAPTER 4
306
u
is a basis element
h
> t.
of weight
W ( 1 ) > t implies h E It+l
But:
and thus the result is true for k = 1. Assume that the result is true for all such products with less than
k
terms
and write u = c1 (g .-1)( g .-I)B % 3
where
(gi-l)
Put r
( g .-1) are adjacent factors. 3
and
v(gi)
=
,S
=
v ( g .) 3
and
consider the identity -1 -1 (gi-l) ( g .-1) = (g .-1) (g.-1) + ( g . g . g.g .-1) 3 3 % 3 Z 3 -1 -1 + (9*g .-1) (9; g j gigj-l) 31-
-1 -1 Setting h = gi g j gigj
it follows that
h E [GrrGsl SO
v(h)
> r+s
and thus
5 Gr+s
t
(g.g.-l) (h-1) E 7r+s+l. 31-
Therefore
u = a ( g .-1)( g .-1)f3 2 3 -
= c1 ( g .-1)(gi-l)
3
Observe that a ( h - 1 ) B at least t.
i s a product of at most
+ a ( h - 1 )B
k-1
(mod It+l)
factors but with total height
Consequently, by induction, ~ . ( h - l ) B is an ?-linear combination
It+lof the h ' s
modulo
B
a ( g .-11 ( g .-1)f3 3
of weight
t.
Thus we are left to verify that
is such an F-linear combination of the
A's,
By the foregoing, in proving the result we may harmlessly interchange adjacent factors.
This implies that we can put all the factors in their natural order.
Thus it suffices to consider u
n
n with
bi 2 0,
u = h(b
1,
...,b
that bi 2 p
Z b. = k
and
and v ( h ) 2 t
for some i.
is a product o f fewer than k so
that
8 E Gsp
C
biv(gi) 2 t.
If b . < p
i=1
2
)
of the form
2
for all
so that the result follows.
Then
(gi-lIp = (9:-1)
terms.
and V ( G 1 2 s p .
Finally, assume
occurs in u
Moreover, if V ( g
.)
z
=
i, then
and
s , then gi E Gs
The conclusion is that in this new
DIMENSION SUBGROUPS
u,
representation of
307
the sum of the heights i s still 2 t .
Hence the required
assertion follows by induction. (iil E
It.
are linearly independent and that w ( A 1 2 t implies
x's
We know that the
repeated application of (il , we see that the set span It modulo
Is+l
for any s 2 t.
for all sufficiently large Since Gd+l
=
of
span I t .
x's
It therefore suffices to verify that these
with s
X'S
Invoking a
> W(x1
Thus we need only show that
2 t
Is+l = O
S.
1, it follows that V ( g )
Q
d
for all g E G,g # 1.
There-
fore in the product
.. (+,-1)
u = (+ -1) (xg-l).
... +
we have V ( x 1 + \)(r21+
dk.
V(Zkl
Hence I d ,
5 I (G)
and since I (G)
is nilpotent by Corollary 3.1.2, the result follows. We are now ready to prove the following result. Let G
1.4. THEOREM. (Jennings(l94111.
G = G > G 1 -
be a finite N -sequence with
P
filtration of I ( G l
be a finite p-group, let
>...>G >G - 1 - d - d+l-
2 -
IGi/Gi+ll
=
i Q d , and let 11.1 z
pei, 1
Put To = FG,
determined by this sequence.
rn
be the
d C (p-l)nen n=l
=
and define the integers cn by
d X(P-lli)ei n ( 1 + ~ i++... ~2i $=1 +
(i) I ,# 0 (iil
but
=
en = diyn/rn+l
(iii) G n (l+Iii) = G.
- m - c en? n=o
0
for all
n E {O,l,
...,ml
for all i E {1,2 ,...,d + l }
Proof. Note that the definition of N -sequence allows some of the e ' s
P
zero, but this does not effect our calculations. any refinement of chosen, then e;
{Gs} i s
{Hsl
observe also that if
into a composition series and if gi E H . - H i + l z
precisely the number of gjs
to be is
is
of height j .
d
Keep the notation of Lemma 1.2 so that the elements (i)
h (al,.
n
.., a , l = n
i=1
a
(gi-l) i
3 08
CHAPTER 4
form an F-basis for FG.
h occurs when all
Then the largest possible wieght of
ai are equal to p-1.
Since there are precisely
e
of the g ! s
j
with height
we have in this case that
j,
d
n
1 # 0 but
Hence, by Lemma 1.3(i),
=
m
0.
(ii) Owing to Lemma 1.3(i), the number f n = d i dn/In+l 1's
number of
of weight n.
coincides with the
On the other hand, the weight of
=
A(a
,...,an )
is
w(X)
a v ( g l ) + a 2 v ( g 2 )+
... + anv(gn)
is equal to the number of ways of choosing 0 4 a .
which shows that f such that
=
But the latter is the coefficient of
V ( h ) = II.
A ( 1 + Xv(g$
2V(P.)
+x
... + x
+
x"
< p-1
in the product
(p-1)v(gil )
i=1
Finally, since there are precisely
m
f,x"
C
=
(iiil Fix i E
n {1,2
S
=
g.
W(g-1) =
follows.
j
we get
If
...,m}.
E {o,l,
g E Gi, then v(g) 3 i so that g-1 E Ti
Conversely, let g E G-G {Gs}
of
i so that V ( g ) < i, and let
into a composition series.
for some j 2 1 and in choosing our generator for H j / H j + l
gj
=
X(p-l)ilei
,...,d+l}.
be a refinement
2
n ( l + x i + ~...* i+ +
fn for all n
=
by definition of Ti. {H }
i
i= 1
m=o
This proves that c
with v(gi)
of the g !S
e
d
Since g f l , g E H j - H j + l we can clearly take
With this choice of generator, g-1 = 9.-1 i s a basis element of weight 3 v(gl C i. Hence, by Lemmas 1 . 2 and 1.3(ii), 9-14 Ti and the result
'
In order to apply the previous result to the powers of the augmentation ideal
of F G ,
we characterize the dimension subgroups in a purely group-theoretic
manner.
Assume that
{Gi] 2
{Gi}
{Hi} if and only if
and G. 2 2 -
1.5. THEOREM. (Jennings (1941)). the filtration of T(G1
{Hi} are two N -sequences in
P H. for all i. 2 Let
determined by
G
G.
Then we write
be a finite p-group and let
IDi CFGI 1.
{Ii) be
DIMENSION SUBGROUPS
ti1
ID ( F G ) } is the unique smallest N -sequence z' P
the sequence
(ii) 1;
=
3 09
i for all .i
I(G1
ti) By Lemma 1.1, the dimension subgroups D . ( F G )
Proof.
Since I ( G )
sequence.
{ G .}
Assume that
P
is an N -sequence
{Di(FG)}.
P
P
P
filtration of
I(G)
-
Then obviously
{Gi n Di ( F G ) 1
It therefore suffices to verify that there is no
{Di ( F G ) 1.
N -sequence smaller than Fix an N -sequence
P
for some integer d .
is nilpotent, Dd+l(PG) = 1
is an arbitrary N -sequence.
form an N
{Gi)
IDi ( F G ) 1,
{Gi}
with
{Gi}.
determined by
and let {Jil
Since I ( G )
=
Jl
be the
we have
On the other hand, let u = ( 3 -1) (cc -1). 1
2
..(zk-l)
be any generating element of S. so that V("
Then
"i
GV(CC.1
C
-
DV(3c.l ( F G )
)
+
V(Cc2)
-I.
... +
V(Z
k1 P i
so that xi - 1 E I ( G 1
VC.qi)
and
z
as required.
i
(ii) By (i1, { I ( G ) } {Gi}
=
{Di(FGl}.
=
{Ji} is the filtration of I ( G 1 i
Thus J .
z
=
1, = T(G1
z
determined by
and the result follows.
The following result provides a more explicit characterization of dimension subgroups.
Although it is valid for an arbitrary group G ,
the special case
below will suffice for our subsequent investigations. 1.6. THEOREM.
Let G
(i) (Jennings (19411)
be a finite p-group.
IG)
Dn(FG) = Mn IP
Then, for all n 2 1,
CHAPTER 4
310
(Lazard (19531 1
(iil
Proof.
Dn(FG1
Lnlp(G1
=
D (FG)
We f i r s t show t h a t
n
n.
t r i v i a l , w e argue by induction on
< n.
is t r u e f o r a l l subscripts
N -sequence of P
G.
2 M,
(GI.
The case
n
= 1 being
IP
n
Assume t h a t
>
and t h a t t h e a s s e r t i o n
2
{D.VG)}
We know, from Lemma 1.1, t h a t
i s an
Since
t h i s yields
Let
i
be minimal with
we have
Dn(FG) 3 DiVGlp Bearing i n mind t h a t derive
Dn(FG)
2 M,
Mn, p
(G)
3.Mi,p(G)P
i s generated by
,GI
[Mn-l,p(G)
and
Mi
(GIp,
we
IP
(GI. I P
We next show t h a t
M (GI n tP
3L
Mi ,P 'Z
which i m p l i e s , by induction on
Taking i n t o account t h a t
Mn, p
(GI
contains a l l
rP
Since
>_ yi ( G ) .
GI
for a l l
Furthermore,
M
iP ,P
CG)
i
3M i , p ( G ) P
j, t h a t j
j
. (GI ip3 , p
2 Mi,p ( G I p 2 yC(GIP
{MnSp(G)1
YC(GIPJ
CG).
1. [Mi,l,p(Gl,
GI
M, -(Gl
it follows by induction t h a t
M
nrP
i s a decreasing sequence, we conclude t h a t
with
ipj 2 n
and hence
M (GI ntp
2 LnIP(G).
We
have t h e r e f o r e shown t h a t
Dn(FG) Thus it s u f f i c e s t o prove t h a t Passman (1977)) t h a t
(GI}
{L
L
n,P
(GI
2 Dn(FG). P
is t h e unique minimal N -sequence of
t h e r e s u l t follows.
I t can be shown (e.g. s e e
i s an N -sequence of
I P
{Dn(,?GIJ
1. M n , p ( G ) 3 L n l P ( G )
P
G.
G. Hence
But, by Theorem 1.5,
(GI
L I P
3 DnIFG)
and
DIMENSION SUBGROUPS
311
We now turn our attention to the powers of the augmentation ideal o f
FG.
Since the corresponding result entails a fair amount of notation, it will be especially useful to assemble most of it in one place.
G is a finite p-group G. =
z
M
G = G
G = H
i,p
(GI
3 G
1 -
3
1
H
IGi/Gi+ll
and
=
pei
3
... 3 Gd 2 Gd+l
=
3
... 3 Hn
= 1
2 2
3
Hn+l
1 G
is a composition series o f
which refines
IGi). gi E H.-H z
V(gil t(G)
i+l’
1 G
i
G n
is the largest
m for which
9;
is the nilpotency index of J(FG1
1.7. THEOREM. (Jennings (1941)).
Grn =
I(G1
With the notation above, the following proper-
ties hold:
z
(i) For all k
the elements
1,
n
n n Qi-ll
U
i with 0
a. c p 2
2=1
uiv(gi) 2 k form an F-basis for TCG)k n i=1 U i with 0 G (ii) For all k 1, the elements Cgi-ll n z=1 k+l C uiv(gil = k form an F-basis for I ( G ) modulo T ( G ) i=1 d (iii) t C G l = 1 + (p-1) C i e d i
n
(iv) Let m
=
C (p-l)z’ei and define the integers C k
i=1
d
n( 1 + ~ i + ~ 2 +i ...
+
X(P-lli)ei
k
Then ck = dimICG) / I ( G )
k+i
for all
k=0 1
c p
and
by
=
i=1
Ui
and
k
p .
k Gm.
F Proof. Let Theorem 1 . 6 C i ) ,
(1.1 be the filtration of I C G ) Z G . = D.(FG) z z
determined by
and so, by Theorem 1.5(ii),
{GiI.
Ii = ICGIi
By
The required assertions now follow by appealing to Lemma 1.3 and Theorem 1.4. As
an application of Theorem 1.7(iv), we prove the following result.
1.8. THEOREM. Let G
and
be a finite p-group and let rn
=
tCG1
- 1.
.
for all i.
Then
CHAPTER 4
312
d Proof.
We keep t h e n o t a t i o n of Theorem 1.7 so t h a t
=
xmnd
...
+
(l+~-i
+
m
c
=
C ~ - l l & ? ~ . and
x-(p-l)i ei 1
i=1
m
=Pc~x” n=o
m
c c m-n x” n=o
=
Comparing c o e f f i c i e n t s , we deduce t h a t c
m
Hence, i f
1
< m-1,
n
= c
m-n
then
On t h e o t h e r hand, by Lemma3.11.2, d i d ( G l m = 1.
Since
I(Glm+l = 0,
we
F conclude t h a t d i d ( G ) O / I ( G ) = dimFG/I(G) = 1 = dimICGlm/l(G)m+l
F
F proving t h e case n = 0. If
1
n
=Gm ,
,
F
then by t h e foregoing
n-1 dimFG/I(Gln = dimFG/ICG1
F
F
+ C did(G1 k/ I ( G)k+l k=1 F n- 1
= dimT(Glm
+ C
k=l
F
dimZ.IGlm-k/l(Glm-k+l
F
= did(G)wn+l
F Therefore
a s asserted.
.
I GI
= dimFG = diml.IGln+ dimICGlm-n+l
F
F
F
Next w e determine t h e l e f t and r i g h t a n n i h i l a t o r s of powers of
1.9. COROLLARY. (Hill (19701). Then
Let
I(G).
G be a f i n i t e p-group and l e t m = t(G) -1.
DrMENSION SUBGROUPS
313
On t h e o t h e r hand, by Theorem 3.3.6,
and dimr(T(G)nl = [ G I
-
F
diraI.(.Gjn
F
Applying Theorem 1.8, we t h e r e f o r e deduce t h a t dimR(TCGlnl = dimrU(Glnl = dimI(G)m-i-l+l
F
F
F
Thus
as a s s e r t e d . W e c l o s e this s e c t i o n by providing some observations on t h e nilpotency index
t(G)
of
JCFG).
1.10.PROPOSITION.
(Wallace (196811.
Let
G be a p-group of order
pn.
Then
n ( p - l ) + l C t CGl G pn By P r o p o s j t i o n 3 . i . 8 ( i ) ,
t(G1
[GI
I n p a r t i c u l a r , i n our c a s e we have
t(Gl
pn.
proof.
f o r an a r b i t r a r y f i n i t e group
G.
I n t h e n o t a t i o n o f Theorem 1.
we have
d d Since
C e . = n,
we have
d
a s required.
1.11. THEOREM. (Wallace ( 1 9 6 8 ) ) . G.
Then
Let
P be a normal p-subgroup of a f i n i t e group
CHAPTER 4
314
Proof. By Proposition l.l(ii1, we have
Set n = t Ip) and m = t G / P )
proving that t (G)
.
Since J V P )
=
I(.P)
we have
< t (P1t (G/P1.
We now prove the second inequality.
Since
[ J CF (C/P) 1
Imp'
# 0
it follows from C21 that there exists
be a transversal for P
Let T
s E FP and ti i
E
T.
in
G
and write
# =
t s + 1 1
Then there exists j E {l,21...ln)
... + tnsn
such that s j
with
9 I(P1,
i.e.
Consider now I ( P I n - l
But
(
c
which, by Lemma3.11.2,is spanned by
Z' y . LFp
Then
y)W # 0 , since
!@
and in this expression of
(
C y)rJ
as a sum of elements whose support lie in
.#P distinct cosets of P
in G1 we have
upon noting that the coefficient of It follows that
1 in
( C a#)( X y ) 9Y=p
is
X a # 0. SEP
315
Remark. p = 2
The bounds given in the above theorem cannot be improved. and
IGI = 4.
If
Indeed, let
P is a subgroup of order 2 , then t(P) =
t(G/P)
2
=
and so 4 = 2.2
If G
3 t(G) 2 2 + 2 - 1
is cyclic then t(G1 = 4 and if G
2. COMPUTATION OF
t(P1
L
z2
x
= 3
z2
FOR SOME INDIVI'DUAL p-GROUPS
then t(G)
J ( F P 1 = I(P1
3.
P
Throughout this section, P denotes a finite p-group and F of characteristic p .
=
an arbitrary field
Our aim is to compute the nilpotency index t ( P )
P.
for some individual p-groups
Mn
=
M
n,P
of
For convenience, we put
(PI
for all n 2 1
so that if
P = M1 -> M 2 -2 . . . z M d > M d + l = 1 and
then by Jennings' formula (Theorem 1.7 (iii)I
Let
A be an abelian subgroup of P, let
that in 2.1. 2 E
P.
is a normal subgroup of
P.
z E
Z(P) be of order p and assume
Then both
Ap and < A p , , >
are normal
With this observation, we now prove
THEOREM. (Motose (197811.
Z(P1 be of order p
Let
A be an abelian subgroup of P, let
and assume that d P
and that P/
is
CHAPTER 4
316
elementary abelian.
p (i) If
b
= (P:A),
P/Ap
Put
pCt
t
t+1
= (Ap : A p
S
= the exponent of
A
( t 2 0)
is elementary abelian, then
t ( P ) = Cb (ii) If
p
and
P/Ap
+
s-1 t C ctp ) ( p - 1 ) t=o
+
1
p,
is not elementary abelian but of exponent
t(P1
=
P/Ap
(iii) If the exponent of
s-1 (b + 1 + C C $ t ) (p-1) t=o
p,
is larger than
1
(3)
then
s-1 C ctp ) (p-1)
+
t ( P 1 = (b+p-l
+
then
+
1
(4)
t=o
Proof.
By hypothesis, P' C < A p I z >
P'
(i) O u r hypotheses ensure that
and so
5A p
= A'.
and
Invoking (5), we there-
fore deduce that
M2
=
M3
Mp+1 = Mp + 2
=
=
... ...
=
M
=
M
= Ap
P
= AP2
P2
- - - - - - - - _ _ _ _ _ _ _
M PS-l+l
=
Hence
9-2 = Cs-2'eps-2+1
=
P
P
s-1
... = eps-Ll = o
= c6-1
and therefore, by (11, we have t(P) = 1
+
@-I)
PS-l C ie
2'= 1
i
= 1
+
5-1
t
( p - 1 ) ( b + C ctp )
t=o
t(P)
COMPUTATION OF
317
as required. (iil
Since
€'/<Ap,.>
Ap C- [
=
p
Hence, in both cases p # 2 (iii) Since
P'
since
p
and
5 <Ap,,>= 9 , M
= 2
M
an application of (51 shows that
.,, = MP =
!?+I
(3) follows from (1).
= 2,
...
=
=
<Ap z > =
M P2
APZ
Hence It is
CHAPTER 4
318
'
Thus (41 follows from 111. 2.2. 2 E
COROLLARY.
[PI
Let
=
pr
and assume that there exists a central element
and an element h E P
P of order p
of order ps
< h , z > 4 P and P / < h p , z > ( i ) If
P/
such that
is elementary abelian
is elementary abelian, then
t ( ~= )ps
t
( r - s )(p-1)
(ii) P / < h p > is not elementary abelian but of exponent p ,
(iii) If the exponent of P/
t (PI Proof.
=
Setting A = < h > ,
is elementary abelian.
is larger than p ,
then
then
p s + (r-atp-1) (p-1)
we have Ap
=
,
d P and P/<Ap,z>
Hence the hypotheses o f Theorem 2.1 are satisfied.
Furthermore, in the notation of Theorem 2.1,
we have
ct
= 1
for all t.
Hence
s-1
(
c
c t p l(p-11
=
pS-1
t=o
and the result follows by appealing to Theorem 2.1. Recall that a group G
i s called rnetacyczic if
group whose factor group i s also cyclic. elements a
and b
where the positive integers m,r,s, Set d = h ,t ) and k = t / d . also assume that t Im THEOREM.
Therefore, G
can be generated by two
with defining relations
am = l , b s
2.3.
G has a cyclic normal sub-
=
at
and b-lab
and t
=
ar
satisfy rs
Then, replacing a
by
l(mod
m ) and
bfItk-1).
ak , i f necessary, we may
.
(Koshttani (1977a1, Motose (1978)).
Let P
be a metacyclic p -
t(P)
COMPUTATION OF
=
Proof.
P
If
if
s < k
if
s > k trivial.
i s c y c l i c , then t h e a s s e r t i o n i s
s 3 1 and
assume t h a t
I
Pn+pS-l n+s-k kP +P
3 19
k
1.
t , t t+l > p =
p2
+ 3p - 3
P = < a , b , c I z = b P = c ? 2 = 1 , a - 1 b a = bcnP,a-'ca=bc,b- 1c b = c > where n
P
=
is a quadratic nonresidue modulo p .
p2+3p-3 =
d,
-1
322
CWTER 4
2 . 6 . TABLE monabelian groups of order
OF ORDER pa
3. CHARACPERIZATION OF GROUPS P t ( P ) =pa-’+p
- l,t(PI = pa-1
Throughout this section, P
AND t[P) =
of J(FP1
WITH
t(P)
=
U(p-1)
U
+ l,t(PI = p ,
(a+l)( p - 1 ) + 1 a
denotes a group of order p ,a 2 1, and F
arbitrary field of characteristic p index t ( P )
24)
3
0.
an
By Proposition 1.8, the nilpotency
satisfies a (p-1)
+1
t (PI
a
p
It i s therefore natural to investigate those groups P t ( P ) = a@-1)
+
1.
for which t ( P ) = pa a If P is an abelian group of order p ,a 2 , then by
Corollary 3 . 2 . 5 , the second highest value of t ( P )
is
pa-’ + p
case P i s not cyclic but has a cyclic subgroup of index p . generalize this fact by characterizing arbitrary groups P for which t ( P 1 = pa-’ groups P
for which
+ p - 1. t(P1 = p
-1
and in this
In what follows we a
of order p ,a 2 2 ,
In the final part of the section we examine those or
t ( P ) = (a+l)(p-1)
+ 1,
the latter being
the second lowest value of t P ) , by virtue of Jennings’ formula (Theorem
1.7 l i i i l
.
or
CHARACTERXZATION OF GROUPS ? .
323
We start by r e c a l l i n g t h e following p i e c e of information.
@(GI
G = <X,g>
g
E
MI
so
@(GI
G/@ GI
ticular, i f
Thus
5M
G'
2
G
Thus
G
G/CJ(GI
x
E G.
p.
has order
CG:@(Gl) = p , then
P
X
M
E
for a l l
.
2 E
M of
G/@(G)
d
G,
.
(G:M) = p.
G as
(G:M)
@(G)
then
p.
5
@(GI
G/@(G).
is a
G.
i s a generating s e t of
may be omitted from any generating s e t of
(G:O(G))
and so
=
p
d
,
G;
Howhence
as asserted..
W e are now ready t o prove
3 . 2 . THEOREM.
blotose and Ninomiya 11975a11.
Let
P be a p-group of o r d e r
pa,a 2 1. (il (iil
t ( P l = a@-1) + 1 i f and only i f
t(P) Proof.
=
pa
(il
i f and only i f For a l l
n
a
1,
P=M 3 M 1 -
e If
IMi/Mi+ll
= p
i,
P i s elementary a b e l i a n
P is c y c l i c Mn = M
(PI and w r i t e n*P
put
3 . . . 3 M
2 -
3 M
- d-
d+l
But
On t h e o t h e r
..
.
2
p
- 1. be positive integers and l e t
are positive integers with
ing conditions are equivalent:
e
1
e
2
... 2 e n'
F =
n C ei i=1
Then the follow-
P
CHARACTERIZATION OF GROWS
n
c
(il
e
-
q
325
r- 1
(n-11 2 q
i=1 (ii) n = 3 , r = 3 , q = 2
or n=2,e
= 1 2
Proof. The implication (ii) Obviously, if x
If n ? 3
>
2
and y
+
>t >
(il 2,
is obvious.
So assume that (i) holds.
then
then by ( 2 ) we have i=1
e +e -1 2 4
1
2
e +.,.+e + q 3 n+qn-2-
2-
n e ~ . q ' + n - l
1=1 e +e -1 e e > q l - q ' - q 2 + ( n - 2 ) q + n - 3 =
Thus n = 3 , e
e
= 2
r
= 3.
(qP- 2
and
= 1
e e -1 (q ' - 4 ) ( q
3
Finally, if n
=
2
-
2 ) (U-1)
-
1 1 + (72-3) ( q i 1 1 2
= 0
o
which implies that q
=
2
and
then e e -1 q - 1 2 ( q I - q ) ( q * -1)
and so e 3.4.
=
1, as required.
THEOREM. (Koshitani (1977a), Motose (19781).
a of order p . (i) t ( P )
=
(ii) t ( P ) (iii) P
+ p -1
pa-1
contains an element of order p
(iii) =* (i): Let h E P
Cii)
.
be of order p
a- 1
.
Then
is normal.
is not cyclic, for otherwise there exists x E P such that
P is cyclic. t ( P ) _pa-1
a-1
(il * (ii): Trivial
Proof.
P/
is noncyclic
Then the following conditions are equivalent:
pa-'
7
Assume that P
+
Hence P / < h p >
9=
h
and
is elementary abelian and, by Corollary 2.2(i),
p - 1.
* (iii): First assume that
NOW
P
is abelian, say P
z Pe
Then, by Corollary 3 . 2 . 5 ,
k=l
1
32h
CHAPTER 4
n C ek and n 2 2. O e n g to Lemma 3.3, (31 holds only for k=1 n = 2,e = a - l , e = 1. n u s P z Z x Z and so P contains an element of 2 a- 1 Pa-l P order p
where a =
.
By the foregoing, we may assume that P is nonabelian so that a
We
If a = 3, then either P contains an element of
argue by induction on a. order p2 or P
3.
M(p).
However, t(Mlp))
=
4p-3
4.
-P
Let z
be a central element of order
is not cyclic and, by Theorem 1.9, t ( F )
x
Hence, by the induction hypothesis, there exists an element
F.
Because
is not cyclic,
implication (iii)
* ti).
normal and P/
is elementary abelian, by virtue of
Thus there exists an element h is elementary abelian.
such that a- 2 k
hp
Now if
On the other hand, if h
is of order pa-’.
then h
p/<xp>
of order pa-2 in
=
z (0 C
is
k < p),
is of order pa-2
then,
by Corollary 2.2, t(P) =pa-2+2(p-1)
or,
t t ~ =pa-2+3(p-l) )
Since in all of these cases *(PI d pa-’,
t ( P ) = pa-2 + p 2 - 1
or
we conclude that h
is of order pa-’,
as required. a be a group of order pa,a 2 1.
Let P
3.5. THEOREM. CMotose (1978)).
Then the
following conditions are equivalent: (1) t ( P ) = pa-’
(ii) P
M(3)
Z x Z x Z
or P
2
2
2
Proof. The implication (iil and Corollary 3.2.5.
P Iz
x
... x
So assume that t(P1
,
Z
Pe1
* (il is a consequence of Corollary 2.4(iii1
then n 2 2
.
pa-
If P is abelian, say
and
7 n
n
pa-’
=
= t(~1 =
e
1p
i
-
(n-1)
i=1
by Corollary 3.2.5. Assume that P exponent p
Hence, by Lemma 3.3, P 2 z is nonabelian of order p3.
and so P
M(p1.
x 2
z
x 2
z
2
.
By Theorem 3.4,
P is of
However, p 2 = t (PI = 4p-3, by Corollary 2.4(iiiI ,
P
CHARACTERIZATION OF GROUPS
and thus p
=
3.
Finally, assume that P be of minimal order with
p
and put
321
=
P/<x>
then, by Theorem 3 . 4 ,
. -P
exists an element h E P
i s
nunaheliad an6 that a 2 4 .
t(P) = pa-1.
Let
t(F)
Then
pa-2
x E P be a central element of order by Theorem 1.9. of order p
contains an element
of order p
a- 1
pa-'
=
t (PI= p
or pa-'
=
+ p -1
a-1
+
pa-2
or pa-l
3 ~ p - 1 1 or
then a-1 2 4
p a-1
and so our choice of P
pa-2
= =
pa-2
and hence
P
2
z2 x z2
F
M(3J
x
z2 ,
or
=
.
=
23.
__-- -
pa-2
2 (p-1)
then t @ )= 5 or 6, by Corollary 2.2.
the assumption that t(P1
F
- 5 X z2
+
ensures that
= 23
then by the abelian case proved above,
-
a- 2
t(p)
a- 2 such that P/<x,hp> or p
Since the above equalities are impossible, we infer that 5,
If
Therefore, by Corollary 2 . 2 ,
elementary abelian.
a 2
We may choose P to
is of exponent p .
or P
(ii) Trivial
=)
.
(i): Apply Corollary 2.2 (ii), (iii)
=)
(ii) * (iii): Put
M,
=
Mn ' P (P) and let d 2 1 and e i >
P = M
3 M
0
be such that
> . . . >-M d -> M d + i = l
2 -
and
Then, by Jennings' formula (Theorem 1.7(iii))
; condition
(ii) implies that
d
Since a
,
Cei
=
we conclude that
i=1
d 2
Hence
e
= 3
e
= 4
... = ed
=
P is elementary abelian.
3
C (i-lle i i=1
0 and e E Thus e
= 1
en M and hence
e
1
= a-1,
=
an
.
so
proving that
(p-1) + 1 Furthermore, M 2
by virtue of (41. Let z
be a generator of M 2 .
assume that the exponent of P
(5)
is a central subgroup of
Then P / < z >
P of order p .
is elementary abelian.
is greater than p .
Finally,
Invoking Corollary 2.2Ciii)
and (51, we derive
(a+p-ll (p-1) + 1 and so p = 2.
= t ( P ) = (a+l)(p-1)
+
1
This completes the proof of the theorem.
4. CHARACTERIZATIONS OF p-GROUPS Throughout this section, F
P WITH t ( P )
7.
is a field of characteristic p > 0
and
t(PI
is
the nilpotency index of J ( F P 1 ,
where P
aim is to characterize those P
for which t ( P )
is a p-group of order
7.
a p ,a 2 1.
By hypothesis, t(P)
Our 3
1.
The case t(P) = 2 being a consequence of a general result, namely Theorem 3.12.4, we may assume that t(P)
2.
All the results presented below are extracted from
Koshitani (1977a1, Motose (19781 and Motose and Ninomiya (1975a) 4.1. PROPOSITTON.
The following conditions are equivalent:
t(P) = 3
(il
(ii)PEz!
PzZ
or
2
3
x Z
2
Proof. That (ii) implies (il is a consequence of Corollary 3.2.5. that t ( P ) = 3.
a = 2 and p
=
must have P follows.
By Proposition 1.8, 3 2, or a = 1 , p = 2.
z!
IPI
or 3
=
1 and hence a
In the latter case t ( P )
Since
4.
> a(p-11 + P
Iz!
=
=
Assume
l,p
=
3, or
2, and so we
implies t ( P ) = 4, the result
'
4.2. PROPOSITION. The following conditions are equivalent: (i) t(P) (ii) P
=
4
is one of the following types:
(a] p = 2 and P (bl p = 2
is cyclic of order 4
and P z z !
2
x z
2
xz!
2
Proof. The implication (ii) * (il follows by virtue of Corollary 3.2.5. Assume that t ( P ) = 4.
or a
=
have
t(P) < 3
By Proposition 1.8, 4
or a = 2,p
1,p = 3 ,
=
2, or a = 3,p = 2.
[PI = 4 or
and thus
a(p-1)
IPI
= 8.
+
1 and hence U=l,p=2,
In the first two cases we
The desired conclusion now
follows by appealing to Corollary 3.2.5 and Corollary 2.4Cil,(ii).
4.3. PROPOSITION. (il (iil
The following conditions are equivalent:
t(P) = 5
P is one of the following types:
(al p = 2
and P z z !
(bl p = 2
and P Z D
Ccl
p = 2
and P E Q
(d) p = 2
and P z z !
4
xz!
2
3 3
2
xz!
2
x z xz! 2
2
'
CWAeTER 4
3 30
p = 3
(el
(f) p = 5
and P E z
3
x z
3
z
and P
Proof. The implication (ii)
* Ci) is a consequence of Corollary
Assume that t ( P ) = 5.
Corollary 2 . 4 ( i ) ,(ii).
and, by Jennings‘ formula p - 1 divides 4. If p = 5 then a = 1 and so P z
a
= 1 implies
t(P1 = 3,
we
z
.
By Proposition 1.8, 5 S a C p - 1 ) + 1 Thus aCp-1)
If p = 3 ,
have p = 3,a = 2
(el
p = 2
and P
Proof.
is elementary abelian of order Z 5
The implication (ii)
*
(i) follows by virtue of Corollary 3.2.5
By Proposition 1.8,
6
> a@-1) +
Table 2.6.
Hence p = 2
divides 5.
and
5.
0
virtue of Table 2.6, Corollary 3.2.5 that p = 2
and a = 5.
a normal subgroup Q
of P
If
If a
4,
then the result is true by
and Corollary 2.4Ci) ,(ii).
of index 2
By Theorem 1 . 9 , t (PI
t(&l< 5.
But
= Z4
1 and, by Jennings’ formula p-1
such that Q
a t (Q) + t (&/PI
-
1
contains an element of =
t (Q) + 1 and hence
which contradicts Propositions 4 . 1 - 4.4.
elementary abelian and the result follows. 4.5.
PROPOSITION. The following conditions are equivalent:
(i) t[P1 = 7
P is one of the following types:
(iil (a)
P
1
Z
Finally, assume
P contains an element of order 4, then there is
order 4.
141
and
Thus
P is
,?
CHARACITRIZATION OF p-GROUPS
(b) P
is elementary abelfan of order
(c) P = z (d)
4
x z
0.
and let F be a field
Then the following conditions are equivalent:
(i) J(FG) = FG*J(FH) (ii) J(FG1 = J(FHl*FG (iiil There exists a normal subgroup N
of
G such that N C_ H and N
has p'-
index in G (ivl
fi
is completely reducible for any irreducible FH-module V
completely reducible for any irreducible FG-module
M.
and MH
is
APPLICATIONS
353
fl and (f),are completely reducible for all irreducible FH-modules V (vil (fll, is completely reducible for a l l irreducible FH-modules V and all
(v)
irreducible FG-modules are H-projective (viil
G
(V
IH
is completely reducible for any irreducible FH-module V
M completely reducible for any FG-module M.
completely reducible implies
(6lBis completely reducible for all irreducible FH-modules
(viii)
and MH
V and H
G.
is of p'-index in
(ix) (MHIC is completely reducible €or any irreducible FG-module M. Proof.
Applying Proposition 3.2 for f = 1 and Theorem 3.7 for e = 1, we
see that the conditions (il ,(it),(iv), (vl ,lvi) , (viil , and (ix) are equivalent. Furthermore, by Lemma 3.13, (vil and lviiil are equivalent.
We are therefore
.
left to verify,that (iii) is equivalent to one of the conditions (il , (iv), (viii) For the sake of clarity, we divide the rest of the proof into two steps.
reducible.
Our aim is to prove that f o r all g J V H ) = FamJ(FL1
Put V (F),,
V, iflIH is completely
Assume that fpr m y irreducible FH-module
Step 1.
=
FH/tJ(FH1.
Then
by hypothesis.
V
where
and
S =
Setting k'
FL/A.
#
Proposition 2.4.3(1),
=
H' n gHil
to be the restriction of
8
5 FH-A
J(FH1
is completely reducible.
5 A.
But
FH n gJ(FH)g-I
is a nilpotent ideal of FL.
A and therefore
=
FH n gJ(FH1g-1
=
Put
Hence by Corollary 2.4.5,
C_ JVL) fl gJ(Fmg-'
to FL, it
and so by
FL n g(ann~1g-l= FH n ~ ( g ~ g -n~gJ(FH)g-' i =
since FH
v'
is completely reducible.
By Theorem l . Z ( i 1 ,
S is completely reducible and thus J(FL1 A
=
G,
i s a completely reducible FH-module, hence so is
fallows from Mackey's decomposition that A = ann(fl
L
E
Hence
J(FL)
354
CHAPTER 5
Replacing
g by
g-l,
it follows by t h e same argument t h a t
FH n
g - l J (FH)g = J (F (H n g-lHg) )
Since
J(FH)
Thus
1 J(FL)
S t e p 2. CompZetion
Of
argue by induction on
L = H (1 gHg-l
that
J(FH) = FH.J(FL), a s a s s e r t e d .
and t h e r e f o r e
t h e proof.
IHI
.
To prove (iii), we
Assume t h a t ( v i i i ) holds.
If
H 4G
then we a r e done with
i s a proper subgroup of
tl
N
=
g E G.
f o r some
H.
Suppose
Since (i)and
( v i i i ) are e q u i v a l e n t , it follows from Step 1 t h a t
Hence, by induction, t h e r e i s a normal subgroup
N
hasp’-index in
G.
Since
L
5H ,
F i n a l l y , assume t h a t (iii)holds. Since
(PIN i s a
N
G such t h a t N
of
Because
N
and
(iii) follows. Denote by
an i r r e d u c i b l e FH-module.
d i r e c t sum of FN-modules of t h e form
completely reducible.
5L
hasp’-index i n both
’Y,g E G,
(FIN i s
H
and
F
a f i e l d of character-
G,
it follows
from t h e equivalence of (i)and ( v i i i ) t h a t
Thus
proving ( i l and hence t h e r e s u l t .
4. p-RADICAL GROUPS
Throughout t h i s s e c t i o n ,
istic
p > 0.
If
G
denotes a f i n i t e group and
X i s a s u b s e t of
FG,
we write
r(X)
=
rG(X) and
p-RADICAL GROUPS
R(X1 = RG(X) As
for the right and left annihilators of
usual, if X is a subset of G, then 'X Let H
4.1. LEMMA.
of FH.
7
(i)
(ivl
Let
be a subgroup of G
fl = g - l I g
and let
?
=
FH,
X in F G ,
respectively.
denotes the sum of elements in X.
and let I
be a nilpotent left ideal
n FGsfl.
9EG
is a nilpotent ideal of FG
If I is"an ideal of
355
then
T=
nPgFG SEG
Proof.
and thus
Fix x
in G.
Then
is an ideal of FG.
It follows that
for any integer m 2 1.
Hence, by induction,
for any integer rn 2 0.
Since
potent ideal of
I is nilpotent, we deduce that
FG.
(ii) By Theorem 3.3.6(iii),
I = kH (21H (I11 which implies that FG.1 = LG(rH(I)FG)
since FG
is a free FH-module.
Hence, if q E G ,
then
i
is a nil-
CHAPTER 5
356
a s asserted.
tiii)
Applying (ii) and Theorem 3.3.6
( i i i ) ,we have
a s desired. (iv)
FG
Owing t o Theorem 3.3.6(vI, ( o r FHI
coincide.
Let
Proof.
(Cx
99
+ )H
=
(il
= LH(H
H
be a subgroup o f
G.
Then
The second e q u a l i t y i s a consequence of t h e f a c t t h a t
Z I = 0 f o r a l l g E G. Since h - 1 E RH(H') h E H gh + I U f l 5 R H ( H 1. But I(H) i s of codimension 1, hence
0 i f and only i f
it follows t h a t
I(H1
Invoking (ii), w e t h e r e f o r e d e r i v e
.
a s required.
4.2. LEMMA.
t h e l e f t and r i g h t a n n i h i l a t o r s of an ideal i n
+1 .
Since
is a f r e e FEi-module, w e conclude t h a t
FG
R (H+) = FGR (H+) = F G * I ( H ) , G H as required. (ii) The proof i s s i m i l a r and t h e r e f o r e w i l l be omitted.
4.3.
(il
LEMMA.
Let
n FG*I(#) SEG
P =
be a Sylow p-subgroup of
n I($)FG SfG
G.
i s a n i l p o t e n t i d e a l of
FG
.
p-RADICAL GROUPS
=
Proof.
&G
uzzl
C
a
fiSy"
= 0 for all
357
z E G
and all S E Syl ( G I )
P
(i) Since I(P) is a nilpotent ideal of
FP, the required assertion
is a consequence of Lemma 4.1. (ii) put X
=
fl
FG.I(fl)
and Y =
SEG
X
n I(@)FG. SEG
Then, by (i),
nFG*I(S)= niT(S)FG
= Y =
S s y l (GI
P
E S y l (GI
P
and so the desired conclusion follows by virtue of Lemma 4.2. Following Motose and Ninomiya (1975b), we say that G subgroup H
of p'-index in G I J ( F G 1 c J(FHIFG.
J(FG) C - J(FH)FG
Note that, by Corollary 2 . 3 ,
is equivalent to J ( F G ) C_ F G . J ( F H 1 ,
Corollary 2.3, if J ( F G ) C J I F H ) F G ,
then H
is p-radicai! if for any
and that, by Lemma 3.12, and
is of pf-index.
4.4. THEOREM. (Motose and Ninomiya (1975b)).
The following conditions are
equivalent :
G is p-radical
(il
(ii) S ( F G )
FG-I(P1
for some (and hence all) P E Syl ( G )
P
r (viii) J ( F G ) = { C z @G
Proof.
I
C z
6 s sg
=
11
nd all S E Syl ( G ) }
g G G
0 for all g
E
P
G
and all S E Syl ( 0 1
P
The equivalence of (iii), (iv), (vii) and (viii) follows from Lemma
4.3. (i) * (ii): J(FG)
If P E Syl (GI then P is of p'-index, so by hypothesis
P
5F G - I ( P ) .
(ii) * (iii): If J ( F G )
5 FG*I(P)
J ( F G ) = J(FG)'
for some P E Syl ( G ) I
P
5FG*I(P)g =
FG*I(Pg1
then for all g E G
CHAPTER 5
358
and hence J(FG1
5
n F G * I (S) E S y l (GI
P
Thus,
=
n F G * I ( S ) , by virtue of Lemma 4.3(1). F s y l (GI
P
H be a subgroup of G of p'-index.
(iii) =* [i): Let
S E syl (G) such that S C_ Ii.
P
By hypothesis, J l F G )
Then there exists
5F G * I ( S ) ,
completely reducible for every irreducible FS-module V ,
fl S($lG,
Since
it follows from Corollary 2.4.5 that
reducible, for every irreducible FS-module J(FH)
5 FH*I(S)
V.
so
rf;
is
by Corollary 2.3.
f
is completely
Hence, by Corollary 2.3,
and therefore, by the implication (ti)
=*
liii)
,
Thus
as required. (iv) * (v1 and (iff)
* (vil : Each of the statements follows from the other by
taking annihilators. As
an application of Theorem 4.4, we derive the following two results.
4.5. COROLLARY.
Let H
be a subgroup of
G
of p'-index.
If G
is p-radical,
then so is H. Proof.
Let S be a Sylow p-subgroup of G with
of the implication (iii)
=*
Hence, by Theorem 4.4, if 4.6. COROLLARY.
Let P
trivial FP-module. reducible.
s 5 H.
Then, by the proof
(1) of Theorem 4.4,
G
is p-radical, then so is H.
be a Sylow p-subgroup of
Then G
G
and let lp
is p-radical if and only if
(I
(Ip)
be the
is completely
p-RADICAL GROUPS
Proof.
If
V
=
FP/I(P),
V
then
l p and, by Proposition 2 . 4 . 3 ( i ) ,
fi 8
Hence
359
FG/FG*I(P) J(FG) c FG'I(P).
i s completely r e d u c i b l e i f and only i f
l a t t e r , by Theorem 4.4, i s equivalent t o
G
being p - r a d i c a l ,
Since t h e
the r e s u l t
a
follows.
The next r e s u l t , with t h e exception of t h e f i r s t property of ( i ) ,i s due t o Khatri (1973). 4.7.
THEOREM.
(il
If
G
N
then so a r e
G/N
i s a p-group and
G/N
(iii) I f
G.
be a normal subgroup of
i s p-radical,
N
(ii) I f
N
Let
Is a p'-group,
G/N,
and
is p-radical,
then
then
G
i s p-radical
G i s p-radical i f and only i f
N is p -
radical. (i) Assume t h a t
Proof.
choose a Sylow p - s u k ~ r o u p Q
5 J(FG1
J(FN1
But
i s p - r a d i c a l and l e t
N
of
with
Q
5P.
P E Syl (G). P
W e may
By P r o p o s i t i o n 3 . 1 . 8 ( i ) ,
and so, by Theorem 4.4,
xQ+ and x ( P - Q ) +
Lemma 4 . 2 ( i ) ,
G
have d i s j o i n t s u p p o r t s , so
z E FN-TIQ).
Thus
S ( F N ) C_ F N ' I C Q )
xQ+ = 0
and t h e r e f o r e , by
and so, by Theorem 4.4,
N
i s p-radical. If
H
=
PN, then H
i s of p'-index J(FG)
Thus, by Corollary 2.3, modules
V.
fl
G,
so by hypothesis
5 FG*J(FHI
i s completely r e d u c i b l e f o r a l l i r r e d u c i b l e FH-
In particular,
i s completely reducible.
in
But
(lH1
H/N
i s completely reducible. i s a Sylow p-subgroup of
G/N
Hence
G/N,
(~H/N) so
G/N
is
p - r a d i c a l by Corollary 4.6.
(ii) Assume t h a t
N
i s a normal p-subgroup of
G such t h a t
G/N
i s p-radical.
360
CHAPTER 5
Since N f P, we have H/N = P/N.
By Corollary 4.6, (lp,NIG'N
is completely
n
reducible.
Hence
(lplu
is completely reducible and therefore, by Corollary
G is p-radical.
4.6,
(iiil Assume that G/N Corollary 4.5.
If G
is a p'-group.
Conversely, suppose that
N is p-radical.
group, there is a P E Syl (G) with P C_ N.
5 FN'ItP).
so
is PI, by
Since G/N
is a p ' -
Owing to Theorem 4.4,
P
J(FN1
is p-radical, then
On the other hand, by Proposition 3.1.8, J ( F G ) = FG'JCFN).
Thus
and so
.
G is p-radical, by virtue of Theorem 4.4.
Turning our attention to Frobenius groups, we now prove the following result. 4.8. THEOREM. UChatri (19731,Motose and Ninomiya (1975bI). Frobenius group with kernel N (il If p (ii) If
divides the order o f
Let G
be a
H.
and complement
N, then G is p-radical
p divides the order of H I then G is p-radical if and only if
so
is
H. Proof.
(i) By Lemma 3.7.1 iv), N
Theozem 4 . 7 ( i ) ,
i s
nilpotent and so
But, by Lemma 3.7.1(iiiI, G,"
2
N is p-radical, by
H is apLgroup.
Hence G
is
.
p-radical, by Theorem 4.7 (ii)
(id.) By Theorem 4.7(il, we need only verify that if 8
G.
If P
is a Sylow p-subgroup of H I
is p-radical, then so is
then J(FH) f F H ' I ( P )
by Theorem 4.4.
By Theorem 3.7.7,
and therefore
JIFG) Sinae
P E Syl ( G )
P
C_ FG*I(P)
we conclude, from Theorem 4.4, that G
4.9. THEOREM. Uhatri (1973)I . if so are
5 FH-I(P1 (N'I
G I and G
If G = G
X 1
G
then G 2
is p-radical. is p-radical if and only
p-RADICAL GROUPS
Proof.
GI
Assume t h a t
G
and
Pi E Syl ( G . 1 , i = 1,2.
P
Then, by Theorem 4.4,
a r e p-radical.
5FGi*I(Pi)
J(FGi) where
361
Since, by Theorem 3.2.3,
2
we have
!thus
G
The converse being a consequence of
is p - r a d i c a l , by Theorem 4.4,
Theorem 4.7(il, the r e s u l t follows. 4.10.
LEMMA.
E
r e s p e c t to Proof.
E
Let
If
J(EG1 C - EG*I(PI
P
(,NI, then by Theorem
E Syl
P
JCEGJ
5E G * I ( P I .
by Corollary 3.1.18.
J(FG) C - E G * I ( P 1 n FG
=
FG*I(P)
N.
p-complement
Then
G
5 F G g I ( P ) , then that J ( E G 1 5 EG.I(P).
So t h e Lemma i s true.
by Lemma 4.2(11.
Let
P E
G be a p - n i l p o t e n t group with a b e l i a n
Let
i s p-radical.
Syl ( G )
Since
F
e2JG
(eiFNIG
F
is algebra-
and Let
P
l = e be a decomposition of
.
groups.
Owing t o Lemma 4 . 1 0 , we may harmlessly assume t h a t
i c a l l y closed,
F.
J(FG)
If
Conversely, assume
'l"heorent. ( T s h u s b a (1986)).
Proof.
i s p - r a d i c a l with
it s u f f i c e s t o v e r i f y t h a t
4.4,
The next r e s u l t provides another c l a s s of p-radt-1
4.11.
G
Then
i f and only i f it is p - r a d i c a l w i t h r e s p e c t t o
JVGI C - F G - I ( P I i f and only i f
Then
F.
be a f i e l d extension of
1
+ e
2
+...+
e n
1 a s a sum of orthogonal p r i m i t i v e idempotents of
is a s p l i t t i n g f i e l d foc i s of dimension
IPI
E",
.
each Now
e.FN 2
G = NP,
theref ore
e.PG = (eiFFJ1 (FP) = eiFP 2-
i s one-dimensional!. so
FG = (FN)( F P )
FN. Hence and
CHAPTER 5
362
It follows that the map
I
FP-
eiFG
x I+ eix
is a surjective homomorphism of right FP-modules of the same F-dimension.
FP 2: eiFG
as FP-modules
and so ezZG*T(P) i s a unique maximal FP-submodule of ePG. s
proper FP-submodule of eiFG,
Thus
(1 G
i
G ?z)
But eiJ(FG) is
hence
eiFG*r(P) 3 J(FG1. and therefore FG*I(P) -
2 eiJ (FG)
i G n)
The desired conclusion now follows by virtue of
Theorem 4.4. The result above need not be true f o r an arbitrary p-nilpotent group
c.
The
following example is due to Saksonov (1971) (see also Motose and Ninomiya (1975b)). Let p = 3
and let F be an arbitrary field of characteristic 3 .
Let G
be
the group of order 24 defined by
G =
be the class sum of the conjugacy class containing g E G. c
=
X
Then we have
x+x3+ y +q+x2y +x3y
+ zzZy + .ZX~+ Z&
=
c 3 = l - c
X
c 2
x
= c
x
and therefore
Hence c
x
+ cz - 1
=
ac+x3+y+xy+x2y+x~y+z+zx2y+~x3+zx3y - 1
i s a central nilpotent element of
cx
+ cz
-
FG and so cx + cz - 1 E J(FG1.
However,
1 does not satisfy condition (viii) of Theorem 4.4, since the sum of
the coefficients of c
X
+
c
Z
-
1 over the coset {x,zx,Z2x} of
is
1 # 0.
P-RADICAL GROUPS
363
G is not p-radical.
Thus
Remark.
According to Tshushima 119861, a p-nilpotent group
G is p-radical if
and only if
10pr(G1 ,Dl n Co
10) = 1
P for any p-subgroup D of G. Let
U and V be any FG-modules.
V, written i ( U , V ) ,
and
Recall the intertwinning nwnber for U
is defined to be
i(U,V) = dim Hom(u,v)
F FG Since
we have
for any FG-modules U,V and
W.
Similarly
and therefore
r
i( @ un, n=l
k
r
@ Vm) =
c
k
c
i(Un,Vm’
n = l m=l
m=l
For the rest of this section, E denotes the algebraic closure of F and
.
{ V l ;. .,Vr 1 a full set of nonisomorphic irreducible EG-modules. module
Given an EG-
M, we write 21
M = C akVk k= 1 if
appears ak
Vk
times as a composition factor of M.
irreducible EG-module, then i(V,V) = 1.
Hence i f
Note that if
V is an
M is completely reducible,
r
c a;. With this information at our disposal, we now prove k=1 r r 4.12. LEMMA. Let M % C akVk and N = C bkVk be two EG-modules. k=1 k=1
then by (11, iCM,M) =
r
(il (ii)
iCM,NI
M
Z akbk k-1
is completely reducible if and only if &(M,M) =
r Z a; k= 1
CHAPTER 5
364
r Proof.
M is irreducible and
so M
3
j
Tf
C ak = 1, then
k=l
...,r}.
for some j E {l,
V
11
C ak. k=1
(il We argue by induction on the value
Hence
Hom(M,N) = HomLM,SocN)
EG
EG
and ttierefore, by (11, P
i ( M , N ) = i(M,BocN) G bj = k=1 “kbk Now assume that
Mo is a submodule of M. 0
-+
By looking at the exact sequence
HomIM/Mo,N) +Hom(M,N)
EG’
--f
Hom(M , N )
EG
EG
we deduce that i(M/Mo,N)
+ C(Mo,N)
(2)
3 C(M,N)
r If Mo is a nontrivial submodule of M with M/M
%
C a{Vk k=1
and
M O
r Z aiv, k=l
then by the induction hypothesis
and
r
But M
r GZZ
k=1
(ai+a{lVk, so it follows from (21, (31 and (4) that r r r
as required. Ciii)
I’f M
is completely reducible, then as has been observed earlisr
r
r .L’IM,M) =
C a;. k=1
Conversely, assume that -f(M,M) = C a; k=1
M/SQCM
=
r
r C
aiVk ,, s o d
k=1
Then, by (il, we have
k C= l afVk
and write
P-RADICAL GROUPS
365
and so applying ( 2 ) , we d e r i v e
r
P
P
Thus
and P
i(socM,M) =
C a% k k k=l
(61
Since P
i(SocM,M) =
i (SocM,SocM)
=
C (a")
k
k=1
a{
it follows from (6) t h a t
ak # 0.
Then
L
a submodule of
M.
and Thus
if
a{ # 0.
a; # 0 and so by (51,
with
SocMf:L c M
i(M/socM,M) # 0.
such t h a t
M/L
..,PI
and hence
3
= af
Syl ( G )
3' k
for a l l Let
{l,.
E
uk
= dimUk,
P
Proof.
..,PI
a = j
a'!
-3
and t h e r e f o r e
0 and
Hence t h e r e e x i s t s
a! 3
such t h a t =
M = SocM,
be a p r o j e c t i v e cover of
Uk = P ( V 1 k
and p u t
then
=
i s isomorphic t o a submodule
a" # 0,
But i f
a{
Now assume t h a t
j e (1,.
4.13. LEMMA.
PE
=
ak
The l a t t e r implies t h a t t h e r e e x i s t s
a'! # 0.
ak
ak
=
Vk, 1
0,
a! # 0 3
a contradiction. a s required.
k G P,
let
Then
E
(IP)'
If
%
c
akVk, then by Lemma 1 . 1 0 . 1 2 ,
k=1
On t h e o t h e r hand, by Theorem 2.4.9,
as r e q u i r e d .
.
W e a r e now ready t o provide a n o t h e r c h a r a c t e r i z a t i o n of p - r a d i c a l groups. 4.14. THEOREM. (Saksonov (19711).
Let
E
be t h e a l g e b r a i c c l o s u r e of
F,
let
CHAPTER 5
366
{ V L ,...,V p } be a full set of nonisomorphic irreducible EG-modules and, for each k E {l,..
.p ) ,
Let P E Syl (GI
let uk = dimP(Vk).
E of double (P,P)-cosets in G.
and let
P
dp be the number
Then
r
with equality if and only if G Proof. (Tsushima (1986) E
=
F.
is p-radical.
.
By Lemma 4.10, we may harmlessly assume that
Then, by Lemmas 4. 2(iil and 4.13,
(lpIG is completely reducible if
and only if
Let T be a set of double coset representatives for
(P,P) in G.
In view of
the above equality, Lemma 4.12(1) and Corollary 4.6, we are left to verify that
BY Theorem 2.4.9 and Mackey decomposition (Theorem 2.6.11, we have Hom(lg, $1
Hom(lpIp, G lp)
FP
FG
P
Horn( @ (1 1 , I ) FPET tPt-lW P @
Hom((1
t€T
FP
P @
P
1 , lP) tPt-hP
Hom
(1
,1
)
t f ~ F (tpt-lw) tpt-lnp tpe-lnp
G G and thus i(lp,lp) = [TI,
4.15. LEMMA.
Let A
proving ( 7 ) .
This completes the proof of the theorem..
be a finite dimensional algebra over a field and let V
a (finitely generated) A-module
I
=
.
be
Then
I$ E End(V) I $ ( V )
5J ( A l V }
A
.
is a nilpotent ideal of End(.V)
A Proof. End(V1. A
It is clear that I is a subspace of the finite-dimensional algebra
If $ E I and I) E End(V),
(4'4) (Vl and
then
A =
$ ( J l ( V l ) C_ $CV) C_ J(A)V
p-RADICAL GROUPS 367
so that $$,$$ E I.
Let n 2 1 be such that
Thus I is an ideal of End(V1. A
Then, for all $ E I
J ( A I n = 0.
and so $n = 0 .
Thus
I is nilpotent, by virtue of Proposition 1.10.5.
The next result is extracted from Tsushima (19861. Let P E Syl (G1
14.16. THEOREM.
End[(l
and let
dP be the number of double (P,P)-
G Then the algebra End[(l ) I P
cosets in G.
,.
P
FG is semisimple if and only if G
is of F-dimension
dp and
is p-radical.
FG Proof.
Put Y
=
and A = End(V1.
Since in the proof of (7) we did
FG not use the assumption that F
is algebraically closed, we have did
F
=
i(l
G G
P’ 1P1 =
dp
Owing to Corollary 4.6, we are left to verify that and only if A If
V
V
is completely reducible if
is semisimple.
is completely reducible, then A
Proposition 1.5.2.
is semisimple by Corollary 1.3.8 and
Conversely, assume that A
is semisimple.
be a full set of nonisomorphic irreducible FG-modules.
k E {l, ...,r } ,
l p i s an irreducible constituent of
from Nakayama reciprocity (Theorem 2.4.101 that each constituent of
V / J ( F G )V .
Let
{Vl,.
. . ,Vr}
Since for all SOC((V~)~), it follows
Vk
is an irreducible
Hence
for some positive integers Assume
by way of contradiction that J(PG1V # 0.
ducible submodule W which is a copy of
Vj
by (81, there is a surjective FG-homomorphism
Then J(FG)V
for some j E (1
has an irre-
,...,r } .
Hence,
368
CHAPTER 5
is a nonzero element of
A
conclude t h a t
A
4CV)
such t h a t
C_J(FGIV.
Invoking Lemma 4 . 1 5 , we
is n o t semisimple, a c o n t r a d i c t i o n .
As an a p p l i c a t i o n of t h e r e s u l t above, we now prove t h e following group-
theoretkc f a c t . 14.17. COROLLARY. (Saksonov (1971)). of
G,
let
P E Syl (GI
P
d
and l e t
r be t h e number of p-regular c l a s s e s
Let
G.
be t h e number of double ( P , P ) - c o s e t s i n
P
Then
dp2 r Proof.
F
Let
by Theorem 2.3.2,
be an a l g e b r a i c a l l y closed f i e l d of c h a r a c t e r i s t i c there e x i s t s exactly
i r r e d u c i b l e FG-modules.
V
say
P,
,..., V p ,
p.
Then,
of nonisomorphic
Keeping t h e n o t a t i o n of Theorem 14.16, w r i t e .r
f3bkVk
SOCV
(9)
k=1 Then we have
bk = i(Vk,S0CV) =
...,r} .
k E {I,
for a l l
irv,,v,
= i(Nk)p, lp)# 0
Since
P
i(V/S(FG)V, SOCV~ =
C akbk k= 1
(11)
we conclude t h a t P
d
= d i d 2
F
Z akbk 2 F , k=1
(12)
a s required. 14.18. of Let
COROLLARY.
G, E
let
P
(Saksonov C1971I1.
E Syl
P
(G) and l e t d,
be t h e a l g e b r a i c closure of
nonisomorphic i r r e d u c i b l e EG-modules. alent:
(iI
dp
= P
Let
r be t h e number of p-regular c l a s s e s
be t h e number of double ( P I P ) - c o s e t s i n F
and l e t
{V
,.. . , V r 1
be a f u l l s e t of
Then t h e following c o n d i t i o n s a r e equiv-
G.
369
P-RADICAL GROUPS
(iiil
G
...,
dinP(Vkl = ] P I for all k E {1,2, r}. E It follows from (lo), Put V = ( l p I G and A = End(V).
is p-radical and
Proof.
(i) * (ii):
FG
A
(111 and (12) that
=
Hom(V/S(FG)V, SocVl
and
FG a k = bk = 1
for all k
It follows from ( E l and (91 that
V/JiFGJ V
E
{1,2,
SocV and hence A
...,rj
End(SocV).
But
FG Socv is completely reducible, so A 14.16,
G is p-radical.
is semisimple and therefore, by Theorem
V
Hence, by Corollary 4.6,
=
SocV and the required
assertion follows by virtue of (91. (ii1 * (iiil:
By Corollary 4.6, G
is p-radical, while by Lemma 4.13,
,...,
r1. dimP(Vkk) = IPI for all k E {1,2 E (iii) =. ( i l : If uk = did'(.Vkl, then (uk/lPI1= 1 for all E Since G is p-radical,
k E 11,...,PI.
by virtue of Theorem 4.14.
Remark.
Owing to Saksonov (19711, all groups satisfying the equivalent con-
ditions of the above corollary are solvable. 14.19. LEMMA.
Let
A
=
Mn(Fl
and let $,$ E A* = Hom(A,F) be such that (A,$)
F and
(A,$]
are syrrrmetric algebras.
Then there exists a nonzero
in F
such
that for all a € 4
$(a1 = hJl(al
Proof.
f
Owing to Lemma 3.3.2(11,
f ( a ) (x) = $(xu)
and
are isomorphisms of (A,A)-bimodules. automorphism.
Let A o
:
A+
A*
g(al (x1
Hence g-lf
be the opposite ring to A
and
g
: A
=
$(xu)
:
A
-+
A
* A* where (x,a
( a @ a )a = a aa 1
2
1
2
A)
is an (A,A)-bimodule
so that A
is an
A @Ao F
module via
E
CHAPTER 5
370
A 8 A" i s a F A i s an i r r e d u c i b l e A 8 A"F such t h a t (g-lf)( a ) = ha f o r
A 8 A"-modules.
g-lf is i n f a c t an automorphism of
Then
F simple algebra with
F
a s a s p l i t t i n g f i e l d and
0#
mod'dle, we conclude t h a t t h e r e e x i s t s all
a
E
A.
Hence
E F
f(a1 = Xg(a) f o r a l l a
E
A.
Since
In p a r t i c u l a r , f o r a l l
a 6 A
we have
a s required. 1 4 . 2 0 . LEMMA.
Let
e s g be an idempotent of
e =
Then, f o r any p-
FG.
SEG singular
ec+ =
o
C e - = tr(.eCfl = 0. SEC f o r any p-singular c E CL(G).
C E CL(G1,
Proof.
n
Choose a p o s i t i v e i n t e g e r Owing t o Lemma 2 . 2 . 1 ,
element.
such t h a t f o r a l l
ePGe
i s simple, then
n g E G , gp
i s a p'-
we have
n n
n e
Furthermore, i f
C e i g p (modIFG,PGl1
= ep
SEG O ur choice of
n ensures t h a t f o r a l l g
C e SEC
Therefore, by Lemma 2.2.2,
Now assume t h a t t h e algebra division ring
D.
If
E
Hence w e may assume t h a t
=
eFGe
0.
C, e
Since
t s t h e c o e f f i c i e n t of
.4
C-l
C
l 0.
and
By
Theorem 2.10,
and hence
If rn
C
ki, then
T(N)i+lP
JIFG)mItN)iP
= annpT(N1
t( N ) 4 - 1 ,
where the latter equality follows by virtue of Corollary 4.1.7, since P over FN.
is free
It therefore follows that
+
I ( N I ~ ( ~ ) - ~ - ~ . S ( F G ) ~ . ~ o( N ) ~ P and thus
Therefore, L ( P ) 2 t ( N )
+
k - 1, where
k = maxRi.
Note that P ( V )
i tive over FH trivial.
and conjugation action o f
on each factor 1(iV)’/I(NIi+’
is
Hence, by Lemma 1.17, @
(P,(V, 8 I w i / I ( N l i + l )
i where X
N
is projec-
H
runs over the FH
@(PH(V,8 X) X
2
F
(and hence FGI-composition factors of FN.
R = m a d (P,(V)
Thus
8 XI
X Since PH(V 8 F
XI
is a summand of P,(V)
8XI
we also have
F L 2 L(PH(V8 X) F
as required.
2.12. THEOREM. (Lorenz (1985)).
Let N
be a normal p-subgroup o f
G, let V
THE LOEWY LENGTH OF PROJECTIVE COVERS
be an irreducible FG-module and assume that G = N H with N
fl H =
1.
For each i
> 0,
let
for some subgroup
Vi denote the FG-module
( I ( N h ( i d + 3Q
v
F where G
.
i+l acts by conjugation on I ( N ) ’ / I ( N )
G (I) Pc(V) ( l H )8 PH(V), where N acts trivially on PH(V). F G (ii) L ( P G ( v ) ) > L(PH(v)) + L ( ( V H ) 1 - 1 G t ( N ) - 1 + maxL(Vi) (iiil L ( ( V H ) i In particular,
G
L((VH)
if and only if all
V.
) =
t(N)
are completely reducible. n
Because X
is projective, it follows that
(ii) If n = L ( p H ( V ))
,
then by (i),
.
proving (ii)
(iii) We first observe that
Thus if m < ni = L ( V i ) ,
then
x 1 PG ( V )
as
required.
395
H of G
CHAPTER 6
396
Here the latter equality follows from Corollary 4.1.7, since
FN.
Therefore, for a l l
i2
(VHIG is free over
0
Since the last assertion is obvious, the result is established.
.
3. THE LOEWY LENGTH OF INDUCED MODULES
Throughout this section, F finite group.
As
denotes a field of characteristic p > 0 and
usual, all modules over a ring R
G a
are assumed to be left and
is an R-module and E = End(W) , then W will also R be regarded as an E-module via If W
finitely generated.
@w 3.1. LEMMA.
Let W
=
Re
=
for all 4 E E , w E W
4tW)
for some idempotent e
of R
and let E
=
End(W). R
Then
W as an E-module is equal to the nilpotency
In particular, the Loewy length of index of J ( E ) . Proof. W
E W.
For each x E @Re, let f
By Proposition 1.5.6,
X
E
E be defined by f x ( w ) = Wx for all
the map
1 , is an anti-isomorphism of rings. J(eRe) = eJ(R)e and so
eRe-
E
5-
j-
Furthermore, by Proposition 1.6.35,
fx E J ( E I i
iv
J (E)
Finally, J(E)'W = 0 i€ and only if the E-module W eRe
i
if and only if x E ( e J ( R ) e )
.
Thus
= Re ( e(8) ~e )
(eJ(R1e)'
=
0.
i s equal to the nilpotency index of
Hence the Loewy length of J(eRe)
=
eJ(R)e.
Since
E", the result follows.
3 . 2 . LEMMA.
Let N
module and let H
be a normal subgroup of
be the inertia group of
V.
G,
let V be an irreducible FNThen each FH-homomorphism
397
THE LOEWY LENGTH OF INDUCED MODULES
€I:
8 I-+
VH
+
8'
VH
extends to a unisue FG-homomorphism 8'
is an F-algebra isomorphism of End(V
H
VG
:
G onto End(V
)
VG
4
.
)
FH
Proof. g l ,...,gk
FG be a left transversal for N in H and
Let g1,g2, ...,gS
a left transversal for N
Then it is immediate that 8' i s Hence the map
in G.
Then
unique element of End(P) FG is an injective homomorphism End(p)
8 I-+ 8'
d
FH algebras. to
V.
Note that
(#)N
and the map
FG
5 $.
is surjective and the result follows.
8.
G
End(V of FG FG I N isomorphic
+
(v
is the sum of all submodules of
Hence for any $ E End(pI, $(#I
extending
This proves that the given map
=
We have at our disposal all the information necessary to prove the following result in which 3.3.
L(p) denotes the Loewy length of the FG-module F.
THEOREM. (Clarke ( 1 9 7 2 ) ) .
Let N be a normal p'-subgroup of
Y and let E
the inertia group of an irreducible FN-module (il J ( F G ) V = FG-J(FHI~IJI = (ii) L ( f i )
=
G, let H be
End(fi).
Then
FG
J(E)~P
for all n
1
is equal to the nilpotency index of J(End(fl) I .
FH Proof. which case
We may take
fi
=
FGe.
V
=
FNe for some primitive idempotent e
Hence, by Lemma 3.1, the Loewy length of
module is equal to the nilpotency index of J ( E 1 .
of as
FN in an E-
Thus (iiI is a consequence of
(i) and Lemma 3.2. Write l = e + e 1
2
+...+
as a sum of primitive idempotents of FN with
e e
rn =
e
.
Then we have
J ( F G ) F = J(FG)e = FG(J(FG)el = FGe(J(FG)e)
+FGe2(J(FG)e)+...+FGem(J(FG)e)
as left FG-modules, where the sum is not necessarily direct.
(1)
For each aEeiFGe,
CHAPTER 6
398
let f a E Hom(FGei,FGe) be defined by f
FG
U
(XI
=
for all x E FGei.
za
Then, as
can be seen from Lemma 1.5.5, the map eiFGe + Horn (FGei ,FGe) FG
is an F-isomorphism. such that f
:
FGe, then there is an a E eiFGe
In particular, if FGei
FGei--+ PGe
is an FG-isomorphism.
For the sake of clarity, we divide the rest of the proof into three steps. Here we prove t h a t
S t e p 1.
J(FG)e = FGeJ(FG)e. it follows from (1) that we need only verify that
Since FGeJ(FG)e cS(FG)e,
eiJ ( F G ) e Let f i
5 eiFGeJ (FG)e
be the block idempotent of FN
Then f;
be the sum of G-conjugates of f i . Now if f
and fi
with eifi i s
(1 G =
e i’ 1 G i G rn,
We may therefore assume that f
1
1
1
f?
and hence
2 1
2 2 2
and let
(2)
a central idempotent of FG.
are not G-conjugate, then f ? f * = 0 eiFGe = e .f .f? F G P f e
i G rn)
= 0
-1
and f i
are G-conjugate, say f , = g fig.
Then -1 (g e i g ) f ,
=
g
-1
(eifilg = 9-le.g z and FNe are in the same block
and so the irreducible FN-modules FN(g-leig)
FNf,.
But N
FN(g-’eig)
is a p‘-group, so FNe FGe
2
FG(g-’eig)
By the foregoing, there is an a E eiFGe isomorphism. y E FGe.
Hence there is a b E eFGei
Therefore sub
=
z
for all
IC
2
and hence
FGei
such that f
:
FGei-
such that fi’(y) in FGei.
Thus
ei = eiab = ( e .a)b = ab and so for any c E eiJ(FG)e ,
c
= e . c = (able = a ( b c ) E eiFGeJ(FG)e 2
This proves (2) and hence the required assertion.
=
FGe
yb
is an FG-
for all
TBE LOEWY LENGTH OF INDUCED MODULES
S t e p 2.
Here we prove that
for a l l
J ( F G l n F = J(E)"#
399
n
>
1.
By Lemma 3.1, it s u f f i c e s t o v e r i f y t h a t
J(FG)ne The case
=
>
for a l l n
( F G e ) (eJ(FG)e)n
n = 1 being proved i n S t e p 1, w e argue by induction on n.
1
So assume
that
k J(FG) e
= ( F G e ) (eJ(FG)e)
J(FG)
Multiplying ( 3 ) on t h e l e f t by
J(FGIk+le
for a l l
k
for a l l
k
n
(3)
n
(4)
gives =
whereas multiplying (3) on t h e r i g h t by
k+l
(J(FG)e) J(FG1e
G
gives
k ( J ( F G ) e ) ( J ( F G ) e ) = ( F G e ) (eJ(FG)eIk+l f o r a l l k 4 n
(5)
Thus we have
J(FG)~+'~= (J(FG)~)~+' = ( J (FG)e )
( J (FG)e )
= (J(FG)"e)
(J(FGle1
( W e ) (eJ(FG)e)n+l
=
proving t h a t (31 holds f o r
S t e p 3.
k
=
(using ( 4 ) with
(using ( 4 ) with
k
=
n)
k - n - 1)
(using (51 with k = n )
n + 1
we nou complete t h e proof by showing that
"fl = FG-J ( F H )"#
J(FG)
We keep t h e n o t a t i o n of Lema 3.2 and p u t
E
= End(#).
for a l l
n 2 1
Then
FH
(by Lemma 3.21
as required.
.
CHAPTER 6
400
Let N
3.4. COROLLARY.
algebraically closed field of characteristic p
let F be an
G,
be a p-nilpotent normal subgroup of
and let V be a principal inde-
n
composable FN-module.
Then
LCFsJ) is equal to
the nilpotency index of
(8) I.
J (End
FG
Our proof of the equality J ( F G ) n F
Proof.
only on the fact that if N
=
J ( E ) n V G in Theorem 3.3 relied
is a pr-group then any two principal indecomposable
FN-modules in the same block are isomorphic.
Since the latter property also
holds under present hypothesis (see Corollary 3.10.101
.
for all n ? 1.
S(FGInf = J ( E I n f l of Lemma 3.1.
3.5. COROLLARY.
deduce that
The desired conclusion now follows by virtue
Further to the assumptions and notation of Theorem 3.3, assume
is an algebraically closed field of characteristic p .
that F exists
, we
Z2(H/N,F*) such that L ( f i 1
c1 E
is equal to the nilpotency index of
Furthermore, if for all q # p
F'(H/NI.
cyclic, then Proof.
L(PI
Then there
the Sylow q-subgroups of H/N
are
is equal to the nilpotency index of F ( H / N ) .
By Theorem 3.4.21,
F"LY/N)
End(#)
for some
c1 E
Z*(H/N,F*).
FH Hence the first assertion follows from Theorem 3.3(ii).
The second assertion is
a
a consequence of the first and Lemma 3.4.9.
G
We close by providing a sufficient condition under which L ( V I the nilpotency index of FP, where P Theorem 3.3.
i s
is equal to
a Sylow p-subgroup of the group H
in
To achieve this, we need to establish some preliminary results
concerning twisted group algebras. 3.6. LEMMa.
Let K
the order of G. Proof. reducible.
be an arbitrary field whose characteristic does not divide
Then, for any c1 E Z 2 ( G , K * ] ,x"lG
is a semisimple K-algebra. V
It suffices to verify that every K'G-module Assume that F/
is a submodule of
V.
over K, its subspace W has a complement in V ,
Since
is completely
v
is a vector space
say
V=W@W'
Let 0
:
V+
W
be the projection map, and let $ : V
-+
V be defined by
THE LOEWY LENGTH OF INDUCED MODULES
u E V and y E G,
Bearing i n mind t h a t f o r a l l
w e deduce t h a t
h'"
i s an
V E
v.
-1 -1 --1 E kr. Then, f o r any z E G , v E W and so v) = z v. -1 z e z v = U and $ ( V l = 0 . S e t t i n g W" = Ker$, it follows t h a t
r
2,
- -
a
If G-submodule of
V
Then, by t h e above,
v
Thus
a
i s a K G-homomorphism.
$
Assume t h a t Accordingly,
W"
=
N
Let
W" n
such t h a t 2,
W"
-$W) E
G
infB(x,y) = 1 f o r a l l
N
3 . 7 . LEMMA.
z,g E N
F i n a l l y , suppose t h a t
V = $(U)
+
( u - $ ( u ) ) EW +PIrr.
T(N)
Let
N
be any f i e l d .
For any
Z2(G,K*) defined by
B(zN,yNl and hence
I(N)
I n what follows we w r i t e
so t h a t
K
and l e t
CinfBl ( z , y l =
KinfB
W = 0.
and so
B E Z2(G/N,K*), l e t infB be t h e element of
identifiable.
e(z
and t h e r e s u l t follows.
be a normal subgroup of
Note t h a t
401
and
KN
{;1I #] M E N}. G,
let
B E Z2(G/N,K*)
a = infa.
(il
KaG-I(N)
(ii) I f
N
is an i d e a l of
KaG
such t h a t
i s a normal p-subgroup of
KO"G.I(N) and t h e nilpotency index of (iii) I f
N
KaG*I(N)
G
KaG/KaG*I(N)
and charK = p,
2
K B (G/N)
then
5 J(KaG) i s equal t o t h a t of
i s a normal Sylow p-subgroup of
are
f o r t h e augmentation i d e a l of
i s t h e K-linear span of be a normal subgroup of
KinfBN
G
J ( f G ) = KaG*I(N)
J(U)
and charK = p ,
then
and l e t
CHAPTER 6
402
In partfcular, by (if), the nilpotency indices of SCpGl
and J(U1 coincide.
(i) Define a surjective K-homomorphism f : KaG
Proof.
B
-+
K (G/N)
by
f(5, = p Then, for all z , y E G ,
(g E G )
we have
is a surjective homomorphism of F-algebras.
Thus f
We are therefore left to
verify that Kerf = f G * I ( N ) . If n
E
N,
n-
then
iE
so I ( N )
Kerf,
be a transversal for N
elements
-
t,t E T.
and let S be the F-linear span of the
G
in
It will next be shown that
K'G
= S
+ KaG*I(Nl
and for this it suffices to verify that each
g
and n E N .
for some t E T
= tn
g=G =
proving that
9E
Fix
Kerf.
5 E
S
+
pG0r[N)
xi
E F,ti E
(6)
gE
S
+ KUG.T(Nl ,g
E G.
Write
Then
a-l(t,nltn
=
a-'(t,n)'i + u-l(t,n~t(n-i),
and hence ( 6 ) is established.
Then, by (61, z 3:
where
and thus'
5 Kerf
A?G*T(NI Let T
5 Kerf
= A
can be written in the form
... + XnTn+y
t1 +
1
T, 1 G i G n , and y E K ' G * I ( N ) .
Because y
E
Kerf,
we
have
f(z) = A T N + 1
which implies that 1 Kerf
K%*I(N)
(ii) Fix n
E
=
... =
A
n
1
... +
= 0 and so
t N
n n
x
E
=
0,
KaG - l ( N ) .
This shows that
and hence that Kerf = K D " G * I ( N ) , as required. N
and write
(N(
=
pd
for some d 2 1.
Since a(z,y) = 1 for
THE LOEWY LENGTH OF INDUCED MODULES
n-
Bearing i n mind t h a t t h e elements we conclude t h a t
I(N)" = 0
KaG , so KaG*I(N)
potency index of
c o n s t i t u t e a K-basis f o r
1,
rn 2 1.
f o r some
= I(N)KO"G
KaG.I(N) c - J(K'G).
Thus
7, n #
403
By (i), KO"G.I(N)
I(N),
i s an i d e a l of
and t h e r e f o r e
Furthermore, t h e above e q u a l i t y shows t h a t t h e n i l -
J ( W ) = I(N) coincides with t h a t of
KaG.I(N).
(iii) D i r e c t consequence of ( i ) ,(ii) and Lemma 3.6.
3.8. PROPOSITION.
Let
N
be a normal p-subgroup of
p
i c a l l y closed f i e l d of c h a r a c t e r i s t i c
6 E Z2(G/N,F*)
(i) There e x i s t s
N
(ii) I f
and l e t
such t h a t
i s a normal Sylow p-subgroup of
c1
G,
c1 €
F
G, l e t
be an algebra-
z2( G , F * ) .
i s cohomologous t o
ir.fB.
then
= F'(G/N)
(a)
F%/J(F"G)
(b)
The nilpotency index of
Proof.
(i) By Lemma 3.4.9,
FaG
i s equal t o t h a t of
J(FN1.
we may harmlessly assume t h a t f o r a l l x,y E N
Fix
g E G,n E N
and w r i t e
pa
=
(NI
.
(7)
Then
----1 = A 9ng-l
gng k E F*
f o r some
and so by (71,
d Hence
hP
= 1,
so
h = 1 and t h u s c--
gng Clearly
F'G.I(NI
of t h e set
1;
i s a l e f t i d e a l of
- iln E N } ,
-
-1 =
gng-l
a
F G.
FaG*I(N1
I(N)
i s t h e F-linear
(8)
span
p G * I ( N ) i s t h e F - l i n e a r span of t h e s e t
(;(;lg I E) G n W e claim t h a t
Because
f o r a l l g E G,n E N
i s an i d e a l .
E NI
To s u b s t a n t i a t e our claim, we need only
404
CHAPTER 6
as claimed.
T be a t r a n s v e r s a l f o r N i n G containing 1, and l e t S be t h e F-
Let
{ilt E TI.
l i n e a r span of
We now prove t h a t
FaG Given
and so
t
E T
and
F'G*II(NI
x,y E N ,
=
S 63 P G * I ( N )
a s F-spaces
(9)
we have
is i n f a c t t h e F-linear span of
Observe t h a t t h e l a t t e r s e t combined with
T c o n s i s t s of
IGI
elements.
The
equality
now proves ( 9 ) , Setting
t N = Z + FaG.I(N),
algebra with t h e elements e x i s t unique
i t follows from (91 t h a t
{ G I t E 7')
t E T , n E N with
a s a basis.
t t = tn. 1
2
Given
Setting
6 ( t l I , t 2 N ) = a ( t l, t 2 ) a - l ( t , n ) we have
and so
FO"G/FO"G.l(N)
i s an
t ,t2E T , t h e r e 1
F-
GROUPS OF P-LENGTH 2
Given Define
g E G,
t h e r e e x t s t unique
h(g)
F* by
h : G-
=
tCg1 E T
405
and
a ( t ( g ), n ( g ) ) , s e t
n(g) E N
y = a(6h)
{ i l g E G}
,
i s an F-basis of
f : F aG
Hence t h e map homomorphism.
+
Y(x,y)
proving t h a t
a = inf8.
3.9.
let
N
Let
$I
G,
f(z) 2
t(g)n(gl.
= h(g)g,g€G.
determines an F-algebra
=
we have
Y
Hence
=
i n f @ a s required.
F be an a l g e b r a i c a l l y closed f i e l d of c h a r a c t e r i s t i c
v.
If
P
G
of
E Syl
P
H
and l e t
be t h e i n e r t i a group of
(HI assume t h a t PN
i s equal t o t h e nilpotency index of
a H.
Then t h e
.
J(FP).
Apply Corollary 3.5 and P r o p o s i t i o n 3.8 ( i i l (bl
4. GROUPS OF p-LENGTH
.
2
G
Throughout t h i s s e c t i o n ,
p.
E
be a normal p'-subgroup
Loewy length of
istic
=
by a cohomologous cocycle, i f necessary, we may assume
c1
an i r r e d u c i b l e FN-module
Proof.
g
Now apply Lemma 3.1.
COROLLARY.
p > 0,
x,y
defined by
B(xN,yN).
=
(ii)By ( i ) ,r e p l a c i n g
that
FB(G/N)
Thus, given
and
-
-g
FYG with x u = Y(x,g)xg and -I
Then
with
denotes a f i n i t e group and
F a f i e l d of character-
All conventions and n o t a t i o n s adopted i n t h e previous s e c t i o n remain
i n force. Our aim i s t o provide circumstances under which t h e i n e q u a l i t y of Theorem One of our r e s u l t s w i l l prove t h a t t h i s i s always
2 . 1 2 ( i i ) becomes an e q u a l i t y . t h e case i f 4.1.
H
LEMMA.
is p - n i l p o t e n t with elementary a b e l i a n Sylow p-subgroups. Let
V and
W
be FG-modules.
=
(J'V/Ji+'V)
xij (0 G
< G n-1,O
G j G
m-l),
Set
n
@ (jW/>+lh')
F where
J = J(FG).
Then
=
L ( V ) ,m = L ( W )
and
CHAPTER 6
406
Proof.
Y.. $3
Put
=
8 $W.
V'J
Then
F
3 Y
-
'ij
.+Yi
i+l,j
,j+l
and
+
Y . ./(Yi+l 23
NOW
Y
let
=
1
C Y.. i + j = i $3
for
R G n+m-l.
0
O = Y
c
C Y
= x.. w
Yi,j+l)
n+m-l - n+w2 -
Then
''*
CY0=V8W F
and t h e canonical map
?
YR
yiji+j=R y i e l d s a s u r j e c t i v e homomorphism
y xij - i +@j = tYzj/(Yi+l,j + Yi,j+ll- xR/xk+l =%
i+j=R where
X
'
=
c
X...
It follows t h a t
i+j=R ' 3
thus proving t h e a s s e r t i o n . 4.2. Let
LEMMA.
Let
N
be a normal subgroup of
W be an FG-module and s e t
V
=
G
WN)
G
such t h a t
W 8 (lNl F
L(V) Proof.
Put
I(M)' = Si*FM,
M
=
where
G/N J
t(G/N1 + L(W)
and view = J(FG1,
FM
ni.
.
G/N
i s a p-group.
Then
1
a s an FG-module v i a
F M S (1 ) N
G
.
Then
and
r(M1i / r ( ~i+l) = ni for suitable integers
-
G
iG
In t h e n o t a t i o n of Lemma 4.1, we t h e r e f o r e have
2
GROUPS OF P-LENGTH
407
a s required. 4.3.
LEMMA.
G
Let
be a p-nilpotent
f i e l d of c h a r a c t e r i s t i c
H
grc p
V
W
and l e t
P
1
fl
L(P(V))
and
=
There e x i s t s a unique i r r e d u c i b l e FG-module Proof.
and
Y
wG
P(V)
That
Y.
3
XN s YN
4.4.
=
t(H/N)
THEOREM.
p > 0 and l e t
has a normal p-subgroup
H
and l e t
P
(il
(iil
P,(VI
G.
of
Q = 0 ,(H), zz
(iiil
If
did,
then
S
then
Let
x
t(H/N)
X such t h a t XN
W.
W.
WG
P(X) 2 P(Y)
Then
X with XN
and
W
Finally, the equality
N
Let
Let
be an a l g e b r a i c a l l y closed f i e l d of
V be an i r r e d u c i b l e FG-module. G
with
W
F
=
NH
N n H
and
W
G
1 f o r some p - n i l p o t e n t
=
be an i r r e d u c i b l e submodule of
T be t h e i n e r t i a group of
Assume t h a t
V Q , where
H.
in
WC
L(PG(V))2
complement i n
vN,
i s a p a r t i c u l a r case of Corollary 3.5.
(Lorenz (1985)).
characteristic
subgroup
i s a submodule of
The e x i s t e n c e of an i r r e d u c i b l e FG-module
follows from Theorem 3.4.5 and Lemma 3.4.9.
L(P(V))
W
i s a p a r t i c u l a r case of Lemma 3 . 4 . 2 6 ( i ) .
be i r r e d u c i b l e FG-modules with
X
hence
be an a l g e b r a i c a l l y closed
be an i r r e d u c i b l e FN-module with i n e r t i a
is an i r r e d u c i b l e FG-module such t h a t
P(m (ii
F
N = 0 ,(GI
where
(i) I f
p
group, l e t
G
t(T/Ql + L ( I V H )
)
-
1 with e q u a l i t y i f
T n S has a normal
S E syl (H)
P
i s elementary a b e l i a n o f o r d e r
n
p
and
pd
is t h e p-part of
F G L ( P G ( V )1 = (n-d) (p-1) + L ( ( V H ) 1
(iv)
If
VQ i s i r r e d u c i b l e , then L(PG(V))2 t ( S )
+ t(N) - 1
and e q u a l i t y holds i f and only if t h e FG-module
is completely reducible.
Here w e view
I(N)i/IUV)i+l
as an FCmodule by l e t t i n g
408
G
CHAPTER 6
a c t by conjugation.
(il
Proof.
and ( i i ) . By Lemma 4 . 3 ( i )
L ( 8 l = t(T/Ql.
Invoking Theorem 2.12,
, we
PH(Y) 2
have
#
and
we i n f e r t h a t
L ( P G ( V l ) 3 t(T/Q) + L ( ( V H ) G )
-
1
and
= (lHfQ 8 =
PG(V,
wc
F
U such t h a t
By Lemma 4 . 3 1 i i l , t h e r e e x i s t s a unique FT-module induced module component.
X
=
U
UH
i s i r r e d u c i b l e and
Assume t h a t
S
C H.
X
Then
-
VH,
H
Q
W.
2
The
s i n c e both have a common FQ'l' nS
i s a normal complement f o r i s normal i n
U
S
in
and p u t
and
= ( uH ) " w x
Y X
X
Thus w e have
I t follows from Lemma 4 . 2 t h a t
LIPG(V1l Since
G/
(iiil
Assume t h a t
p a r t of
dimV.
2
T/Q,
5'
t(G/)
+ LC(VH)G l
-
1
t h e required a s s e r t i o n follows.
is elementary a b e l i a n of o r d e r
By assumption on
S, T
pn
and l e t
pd
be t h e p-
n S has a normal complement i n S,
and
F by Corollary 3 . 2 . 5 ,
t(!P/QI where
pk = IT n S I .
k =n -d
(ivl
By ( i i ) ,w e have
k(p-1)
+
1
U be a s i n t h e proof of (ii)so t h a t
Let
i s n o t d i v i s i b l e by
dimU F Thus
=
p
and hence t h e p-part of
dimV
F
.
which proves t h e a s s e r t i o n by applying (ii)
equals
U
H
VH.
Then
p d = IH/Tl.
409
GROUPS OF P-LENGTH 2
L(PG(V)l
V
Let N
Let n
proof.
This completes the proof of the theorem.
of G.
FG
:
n(y)
NH and N n H
= 1
Then Kern =FG.I(N)
J(FH) = n(J(FG)I .
5 J(FG).
Thus
2
-
for some
FG-I(N) 5 J(FG1 and therefore x
E
Z
E
,TI
Further-
Now, if x € J(FG),
y E FG*I(N) and so
Conversely, let x E FG*I(NI + J(FH1.
and hence r(z] = n ( Z l
some y E J ( F H ) X- z E
we have FG*I(N)
for some y € JtFH1.
r € FG.I(N) + J ( F H ) .
=
be the natural projection.
and, by Lemma l.l(iii),
more, by Proposition 3.l.l(il, =
G with G
Then
FH
-+
is the identity map on FH
then n ( x ] = y
1
(
be a normal p-subgroup of
for some subgroup H
-
L ( ( V H )G
6l3 [I(N)i/I(N)i+ll i20
@
F
4.5. LEMMA.
+
t(S)
G L((VH) 1 2 t (N) with equality if and only if
and, by Theorem 2.12(iii),
is completely reducible.
=
Then
J(FG1.
T(X) =
y
for
Hence
J(FG).
We are now ready to establish the final result of this section. 4.6. THEOREM. CLorenz (19851).
G
=
NH and N n H
complement S ,
=
G with
be a normal p-subgroup of
for some Frobenius group H
1
where
Let N
Q and S are p '
with kernel Q
and
and p-groups, respectively.
Then the
following conditions are equivalent: (i)
t(G1
(iil
=
t(S)
+ t(N1
-
1
6l3 (I(N)i/I(N)i+l) is a completely reducible FG-module, where G acts by
$20 conjugation.
(iii)
C xqs +Q
Proof.
- E xq
E I(N)'"
for all i 2 0, all x
E
I(N)'
and all s E S.
CFQ
(i) * lii):
algebraically closed.
By Corollary 3.1.18, we may harmlessly assume that F By taking
v=
lG
t(G) 2 L(PG(V1l and so L(PG(V)) = t ( S )
+ t(N) -
1.
(ii) * (iii): owing to Theorem 3.7.7,
in Theoreq 4.4, we have t(S)
+ t(N) -
1
Now apply Theorem 4.4(iv).
our assumption on H
implies that
is
410
CHAPTER 6
S(FH) = I ( S ) e where
c
e
q
= 1~1-l
SfQ Set X
FG*I(N) and Y
=
= I(S)e.
Then, by C1) and Lemma 4.5.
S(FG)
=
X+Y
(2)
i+l By hypothesis, J(FG) annihilates each I(N)i/l(N) .
i
I(N)i/I(N)i+l,so Y annihilates each I ( N )
.
x
But
Therefore
also annihilates
(S-1) 1 4
annihi-
SfQ lates J(N)i/l(N)i+l
for all i 2 0 (s-1)(
x
and all s
c
q)z =
- c
zqs
*Q
+Q
Since for any z E I(N)
S.
E
zq
SfQ
the required assertion follows. (iii) * (i): An easy calculation shows that for all i 2 0 e * I ( S ). 1 ( 8 ) i e Set 8
=
t(N)
+
t(S)
-
1.
be written as a product of then a
=
+
eI(N)’+le
=
e * l ( N ) i l ( S ) e+ eI(NIi*le
In view of ( 2 1 , we have to show that if c1 E FG
G
factors each of which belongs to either
involved in a.
t(S). of c1 = 0 .
kz
X
can or
Y
0.
We argue by descending induction on the number
that
(3)
C
t(N1.
If
&,
2 t(N1,
then
c1 E
Xt(N’
,k
= kx(C1l
= 0.
Then the number of factors from Y
of factors from
x
We may therefore assume involved in
c1
is at least
Let n
= n (a) denote the length of the longest consecutive subproduct Y Y Clearly, if n 2 t ( S ) then consisting entirely from factors in Y .
Y
So
belongs to
assume that n < t ( S ) . Then a contains a subproduct which either an Y n YX’Y or to Y ’XiY(i > 0 ) . We examine the first case, the second
being entirely analogous.
Now
n
Thus we have c1 =
c1 1
+
c1 2
with c1 ,a2 E J(FG)
9“
,
but
kz(a2) > kx(u)
and
411
GROUPS OF P-LENGTH 2
Q (a) = Q (a) ,n (a1 > n (a). By induction, we deduce that a = a
x
x 1
hence
a
= 0.
Y 1
Y
This proves the theorem.
1
2
=
0
and
This Page Intentionally Left Blank
413
7 The nilpotency index Let G be a finite group, let F be a field of characteristic p be the nilpotency index of J ( F G ) .
Our aim in this chapter is to provide some
important information on the number t ( G ) . bounds for t ( G ) ,
and let t ( G )
After examining an upper and lower
we investigate circumstances under which the equalities hold.
Almost all of the results presented pertain to p-solvable groups since virtually A number of interesting examples is also
nothing is known in the general case. provided. 1. SOME RESULTS ON p-SOLVABLE GROUPS
Throughout this section, G denotes a finite group and p write
GL(n,p)
and S L ( n , p )
group of degree n
a fixed prime.
We
for the general linear group and the special linear
over the field P
P
of p-elements, respectively.
Our aim
is to record some group-theoretic results required for subsequent investigations. First we record the following conventions. of
If M
5N
are normal subgroups
G, we set
so that
‘G/M (N/W Expressed otherwise, CG(N/MI where the image of g E G
Let @
and consider
V
p
=
CG(N/M)/ M
is the kernel of the homomorphism
is the automorphism of N/M
nM 1.1. LEMMA.
=
G
-+
Aut(N/M),
given by
W g-lngM
be the Frattini subgroup of a p-group
G,
let
(G:@) = p
G / @ to be a d-dimensional vector space over the field
elements (see Lemma 4.3.1).
Then
L?
P
d
of
414
(il
CHAPTER 7
C
= i f and o n l y i f G = < z ,x
particular, i f
1
,...,x > ,
@zl,@x2,...,0xn. In
i s spanned by
n 2 d.
then
G can be generated by e x a c t l y d elements.
(ii) The group
{x1 ' z 2 ,...,xd proof.
2
V
G
generates
(i) If
,...,Oxd 1
{ax
i f and only i f
G = <x ,x2,...,zn>, then
V
The subset
i s generated by
Ox l , . . . , @ x n .
d
elements, w e
Since a d-dimensional v e c t o r space cannot be spanned by less than
X
=
> d.
n
have
.
{xl,. . ,x 1 ,
G = <X,O>,
we have
{Oxl
,...,Ozd 1
Conversely, i f
G = <x
@z,...,Oxd
t h i s case As
1.2.
Let
and
,...,xd > ,
@
t i ) we get
then by
G/O
can be excluded
G=
,...,Xd>.
(Z
,...,O X d > .
V =
In
v. '
AuttG)
and t h e f a c t o r group
d
I
divides
pm
V = C/O. of
U
G
Since
@
Aut(G)/I
Thus
.
n (~'-1) i=l
i s a c h a r a c t e r i s t i c subgroup of
induces an automorphism
i s obviously a homomorphism f o r
d e f i n i t i o n , we have (ii) Fix a b a s i s
( y I , y ,,..., y d ) - -
Clearly we g e t
Oxi
G
- d(d+ll/2.
every automorphism
=
t h e s e t of a l l automorphisms of
I i s a p-group of order l e s s than p 'n-d'd.
( i ) Put
f(U1
and l e t
GL(d,p).
IAut(G1
Proof.
G,
invariant.
i s a normal subgroup of
The subgroup
rn = nd
I
Denote by
i s isomorphic t o a subgroup of
$(@xi)
then by ( i ) ,w e have
be t h e F r a t t i n i subgroup of a p-group
IGI = p".
I
(i) The set
all
V,
i s a b a s i s of
which leave every element of
U-
O
an a p p l i c a t i o n , we prove t h e following r e s u l t of P. H a l l (1933).
(G:O) = pd
where
since
a r e l i n e a r l y independent and so form a b a s i s of
1
THEOREM.
(iil
G = <X>
so
Setting
G.
from any generating s e t of (ii) I f
.. .,OXn > .
=
Since S
1 1 1 divides
(11; hence
by Lemma is an arbi-
1x1.
This shows
that I is a p-group whose order is a divisor of p (n-d)d d(d-1)/2 is a consequence of the fact that IGL(d,p1 I = p
The last statement n lz (pi-l).
i=1
For future use, we need an important property of the upper p-series of a psolvable group, known as the Hall-Higman's lemma (Hall and Higman (1956)).
This
is recorded in 1.3. LEMMA.
Let G be a p-solvable group and let
.
,
subgroup of Op ,p(GI/Opt (G)
@/o ,(GI P
be the Frattini
Then
In particular,
(0 (G11 C_ 0 (GI if 0 ,(GI = 1
(bl
C
(c)
The group G/O
P
G P
P'IP
P (G) is isomorphic to a subgroup of Aut(0
P'rP
(G)/@)
and 4ut(0
(G)/@)
P'SP
0
P'rP
Put K = C (0
(GI C_ K.
G P'tP
(GI/@) and observe that, by Lemma 4.3.1,
By the definition of
p-group, so that, if K
3
0
P'rP
0
(G) of order prime to p .
0
LG)/Op,(G1
P'PP
P'IP
GL(d,p)
(GI/@I
where pd = Proof.
2
(GI, K
0
P'rP
P'rP
(GI cannot be a nontrivial
must contain an element g
Then g
of order prime to p.
(G), K / O
not in
induces an automorphism of But an automorphism of a p-group which is
the identity modulo the Frattini subgroup has order a power of p (Theorem l.Z(ii)). Hence g
induces in 0
P'rP
(G)/@ an automorphism which is not the identity,
contrary to the definition o f
K.
Thus K = 0
P'rP
(GI, as required.
416
CHAPTER 7
W e c l o s e t h i s s e c t i o n by recording some standard group-theoretic f a c t s required f o r subsequent i n v e s t i g a t i o n s .
The p r o o f s of t h e s e f a c t s a r e r a t h e r
involved and t h e r e f o r e w i l l be omitted. I t w i l l be convenient t o introduce t h e following terminology.
i s invoZved i n
K
a group
H
of a subgroup 1.4. THEOREM.
p
= 3
provided
P
K
i s isomorphic t o a homomorphic image
G.
of
G
Let
p
be a p-solvable group and assume t h a t e i t h e r
G,
i s a Sylow p-subgroup of
5
or
G.
SL(2,3) is not involved i n
and
(i) I f
G
W e say t h a t
P
then every normal a b e l i a n subgroup of
Op, ,p(G).
is contained i n
G
(ii) I f a Sylow p-subgroup of
i s of c l a s s a t most
2,
G
then
has p-length
1.
Proof.
( i i l i s an immediate consequence of ( i l .
p,
For a given prime
z
P
x z!
and
P
(il
abelian of o r d e r
we d e f i n e
p
If
p 3 and exponent G
Property
t o be t h e semidirect product of SL(2,p) on z!
P
i s odd, then t h e Sylow p-subgroups of
G = Q d ( 3 ) , then
Proof.
Qd(p)
w i t h r e s p e c t t o t h e n a t u r a l a c t i o n of
SL(2,pl
1.5. THEOREM.
(ii) I f
.
For t h e proof of (i) r e f e r t o Gorenstein (1968,p.2341.
&d(pl
Let
1.6. THEOREM.
i s of 3-length 2.
CG(P n 0 (GI P'rP
II (GI P
Proof.
5 Opr,p (GI
,
Z (P) 5 Op ,p (GI
and, i n p a r t i c u l a r ,
See Gor n s t e i n (1968,p.228) Let
Sylow p-subgroup
G.
P If
p
.
A
and Huppert (1967,p.691)
be an odd prime, and l e t
such t h a t
P of c l a s s
G.
C(P)
1.7. THEOREM.
involved i n
be t h e p-length of
P
and
(see a l s o Koshitani (1982a,p.33)).
G be a p-solvable group with Sylow p-subgroup
II (GI
and l e t
(ii)
a r e non-
For t h e proof of ( i l , r e f e r t o Glauberman (1971, Example 1 1 . 4 ) .
Gorenstein (1968,p.203)
(il
zp.
p.
Property (ii) is an easy e x e r c i s e based on Glauberman (1971,pp.32-331
c(P)
x
0 (GI # 1, G P
G
be a f i n i t e group with a
i s p-solvable and
i s an a b e l i a n normal subgroup
P,
then
Qd(p) is not A
20
P ' IP
(G).
SOME RESULTS ON p-SOLVABLE GROUPS
417
See Glauberman (1986, Lemma 6.31 and Gorenstein (1986, Theorem 8.1.3)
Proof.
( f o r a s h o r t proof r e f e r a l s o t o Koshitani (1982a, Lemma 2 ) .
1.8. THEOREM.
(Hall-Higman).
p.
(il
I? = p
n
P
g be an element of
Let
g
nomial of
in
,
be a p-solvable group of l i n e a r transform-
0 (G) = 1 a c t i n g on a v e c t o r space over a f i e l d
a t i o n s i n which
teristic
G
Let
V
(X- U p ,
is
G
of o r d e r
F
of charac-
Then t h e minimal poly-
where e i t h e r
or
n
2
Mn(pl
Q for
1.10. THEOREM.
D
and
n
i s defined f o r
n2 3
P
Let
i=1 of G/A.
P..
h a s a complement i n a l l
n
b 4
p = 2
if
p
i s odd,
p = 2
p
= 2
P is
and and
n
for
2n
.,Q
.
and
n
>
3
n > 3,
let
2n.
S ( n ) be
defined by
p
n
which c o n t a i n s a
Then
isomorphic t o
= 3,
n > 3,
Finally, for
be a nonabelian p-group of o r d e r
n- 1 c y c l i c subgroup o f o r d e r p
(iii) I f
A
s i x npi ’,
be t h e d i h e d r a l and g e n e r a l i z e d quaternion groups of o r d e r
t h e semidihedral group of o r d e r
(ii) I f
2
=
as follows:
and
(i) I f
be a Sylow p.-subgroup
Pi/A
i f and only i f
IG/Al
with
See Huppert (1967, p. 124).
Recall t h a t t h e group
if
G
be an a b e l i a n normal subgroup of
facts.
then then
P
Mn(P)
i s isomorphic t o
D3
or Q3
P i s isomorphic t o M n ( 2 ) , Dn , Qn or S(n)
CHAPTER 7
418
Proof.
i s s a i d t o be regular, i f f o r any
P
A p-group
x1 ,x2,... ,xn i n
1.11. THEOREM.
<x,y
>'
P
Proof.
i s of exponent
and
P
p,
Throughout t h i s s e c t i o n ,
B
P
then
p > 0.
i s a block of
FG,
G
i s regular
P
then
i s regular
.
i s r e g u l a r , then
P
is abelian.
.
t
2. UPPER AND LOWER BOUNDS FOR
p,
i s l e s s than
P
See Huppert (1967, p.3221.
characteristic
there e x i s t
P be a p-group.
Let
p = 2
(iii) I f
x,y E P
such t h a t
( i ) I f t h e nilpotency c l a s s of
(ii) I f
m
See Gorenstein (1968, g.1931.
F
denotes a f i n i t e group and
As u s u a l ,
t(G1
then we w r i t e
an a r b i t r a r y f i e l d of
i s t h e nilpotency index of
t(B)
J(FG).
f o r t h e nilpotency index of
If
J(B).
We s t a r t by recording some p r o p e r t i e s which a r e immediate consequences of previously e s t a b l i s h e d r e s u l t s .
2.1.
PROPOSITION. (Passman (1970a), Tshushima (1967), Wallace (1968) 1 .
be a p-solvable group of o r d e r
Proof.
2.2.
p rn,
Apply Proposition 3.1.9 Let
LEMMA.
group.
a
B
where
p 1, m and a >
and Corollary 6.2.8.
be t h e p r i n c i p a l block of
FG,
1.
Let
G
Then
.
where
G
i s a p-solvable
Then
t ( B 1 = tCG;/O , ( G I ) P Proof.
D i r e c t consequence of P r o p o s i t i o n 3.1.20.
.
The f i r s t e q u a l i t y i n t h e r e s u l t below i s contained i n a work of Clarke (1972). 2.3.
PROPOSITION.
and l e t
B
Let
G be a p-solvable group of p-length 1, l e t P E Syl ( G )
P
be t h e p r i n c i p a l block of
FG.
Then
t(G1 = t ( P ) = t ( B )
UPPER AND LOWER BOUNDS FOR $(GI
Proof. p-length
419
0 , ( G ) P / O ,(GI E Syl ( C / O , ( G I I
Note that P
P
P
P
and G/O , ( G I
P
is of
P
Hence, by Lemma 2.2, it suffices to prove that * ( G I = t ( P ) .
1.
By
Corollary 3.1.18, we may harmlessly assume that F is algebraically closed. Furthermore, by Proposition 3.1.8(ii), we may also assume that G
is p-nilpotent.
The desired assertion is now a consequence of Morita's result (Corollary 3.10.10). An alternative proof, which avoids the use of Proposition 3.1.8(ii) and Corollary
m
3.10.10, can be obtained by applying Corollary 6.3.9.
2.4. PROPOSITION. (Koshitani (1977b), Tsushima (1978bII.
a
group of order p m , and only if P Proof.
and let P E Syl (GI.
i s cyclic.
t(G)
Then, by Proposition 3.16.20,
Hence, by Proposition 2.3,
t ( C 1 = pa.
Conversely, suppose that t ( G I and @/M
Then
P
=
pa if
is cyclic.
Assume that P
p-length 1. 4.3.2,
where p ) m ,
Let G be a p-solvable
=
pU
.
the Frattini subgroup of H/M.
a-r J (FGI
t ( G ) = t(P1
Put
M
and therefore, by Theorem
o , ( G I ,H P
=
is of
G
=
o ~ , , ~ ( G,pr ) =
IH/MI
Then, by Proposition 3.1.9,
5 FG* J (FH)
and so
Hence, by Propositions 2.1 and 2.3,
t(H)
=
pr , it follows from Theorem 4.3.2 that H/M H/O
is cyclic of order p.
is a subgroup of Aut(H/@I. length 1.
Hence P
H/M
Since H/M
1 Aut t H / G ) I
Thus G/H
is a pr-group,proving that G
=
is of order
Hence, by Lemma 4.3.1,
Therefore
p - 1 and, by Lemma 1.3, G/H is of p -
is cyclic as asserted. a
am, u
p-leng.th 1 and o f order p
if the Sylow p-subgroups of If
= p ,
is cyclic.
2.5. PROPOSITION. (Motose and Ninomiya (1975aI).
Proof.
r
t(H/M1
P E Syl ( G I , P
G
>
1, p ) m.
Let G
be a p-solvable group of
Then t(G) = a(p-1) + 1 if and only
are elementary abelian. then t ( G 1 = t ( P )
by Proposition 2.3.
Now apply
Theorem 4.3.2. 2.6. PROPOSITION. (Motose and Ninomiya (1975a1, Wallace (1968)).
Let G
be a
420
CHAPTER 7
U
p-solvable group of order p m , a 2 1, p
1 m.
Then the following conditions are
equivalent: t(G1 = 3
(il
(ii) pa = 3
Proof.
or pa = 4 and the Sylow 2-subgroups of (i)
(iil
:
Let P
C
are elementary abelian.
be a Sylow p-subgroup of G.
By Proposition
1.1,
and p = 2.
This implies that a = 1 and p = 3 or that a = 2 G
since P
is of p-length 1
t(G) = tlP)
In both cases
is abelian (see Proposition 3.16.20). Thus, in case p
by Proposition 2.3.
U
= 4
,
13
Hence
is elementary
abelian by Proposition 2.4. (iii) =. (il : By the foregoing, G
fs of p-length
Now apply Propositions
1.
2.5 and 2.4,
Let G be a finite group of order Pam, a ? 1, p solvable.
k
m, not necessarily p -
It will next be shown that the inequalities
+
t ( G 1 P a(p-11
1 and
given in Proposition 2.1, no longer hold.
t IG)
pa
In order to accomplish this, we first
prove tHo auxiliary assertions. 2.7. LEMMA.
idempotents
Let 1 of
FG.
=
e +
... + e n ,
t ( G 1 = max{R
Since FG = F G e l @
respectively.
, . . . , & , I < maxb
... @ FGe, 1
Since kZ
< Ti
be the Loewy
Then
... T,I
... @ J ( F G )k en
k
,.. .,&,I.
and ri
and J ( F G e i l = J ( F G ) e i
J ( F G ~= ~J V G ) e t~ Hence t [ G ) = max{k
let Ri
For each i E { 1 , 2 , . . . , ? 2 1 ,
and composition lengths of F G e Z ,
Proof.
where the eZ' are primitive orthogonal
for all
,
we have
for all k 2 1
i, the assertion
follows. 2.8. LEMMA.
Let C = (c..) , 1 6 i , j 6 k ,
be the Cartan matrix of FG.
ZJ
k t(G)
max
I
C c..}
lea 3'11 zJ
Then
UPPER AND LOWER BOUNDS FOR
...,FGek
proof.
Let FGel,FGe2,
FG-modules.
t(G)
421
be all nenisomorphic principal indecomposable
Then, by the definition of
C, cij
is the multiplicity of the irre-
FGei,
as a composition factor of ducible FG-module FGe./J(FG)e 3 j Hence, the composition length of FGei is equal to
Now apply Lemma 2.7.
il + c i 2 +
.
c
2.9. EXAMPLE. (Wallace (1968)).
p
1 m,
i
G k.
... + c ik
There exists a group
of order pam, a
G
1,
such that t ( G 1 < a(p-ll
Let G = A
Proof. Since
1 4
,
5
+ 1
the alternating group of degree 5 , and let p
= 5.
I G ( = 60, we have
a(p-ll
+
It will be shown that t ( G ) 4 4. ally closed.
1 = l(5-11
+
1 = 5
We may harmlessly assume that
F is algebraic-
Then it is well known (see Brauer and Nesbitt (1941)) that the
Cartan matrix of FG is
Invoking Lemma 2.8, we conclude that t(G) G 4 as required. 2.10. EXAMPLE. and let F
Let q
be a power of a prime with q
be a field of characteristic.2.
(i) The Sylow 2-subgroups of G (ii)
G.
l(mod 41, let G-PSL(2,q)
Then for some a
are dihedral of order '2
2
t ( G ) > 2a
In particular (by Proposition 2.11, of
.
t(G)
3
t(P)
where
P
is a Sylow 2-subgroup
Indeed, property (i) is a consequence of a general group-theoretic fact
(see Gorenstein (1968),p.418). ically closed.
To prove (ii), we may assume that F
Owing to Erdmann (1977, Theorem 2(a),p.667),
projective indecomposable module Loewy length L ( V )
of
V
V
there is a
in the principal block of FG
is given by
is algebra-
such that the
422
Hence, by Lemma 2.7
as required. Our next aim is to prove that t ( G 1 G dimJCFGl
1
f.
F and to provide necessary and sufficient conditions under which the equality holds. In what follows, we choose a complete set FGe
1
...,FGek
,FGe 2 ,
of nonisomorphic principal indecomposable FG-modules, where trivial FG-module.
C = Cc..), 1 G i , j G k , 23
We denote by
FGel/JCFG)el is a the Cartan matrix of
FG . 2.11.
With the notation above,
LEMMA.
k t ( G ) G max { Z c . . } G max i < j~ = 1 'J i
+
{dimJ(FG)ei
~
F~
1) G dimJ(FG)
k
+
1
F
The inequality on the left was established in Lemma 2.8.
Proof.
k C c..
j=1 '3
is equal to the composition length of N e i ,
we have
k
proving that
k max { C c , . ) 9max{dW(FG)ei 1 ~ j=1 q '3 F
+
11
The inequality on the right being obvious, the result follows. 2.12.
G.
THEOREM. (Motose and Ninomiya (1975a)).
.
Let P be a Sylow p-subgroup of
Then the following conditions are equivalent:
(i) t(G1
=
dimJ(FG)
+ 1
F (ii) G = P is cyclic or Proof.
(i) * (ii):
1
+
G is a Frobenius group with cyclic complement P.
Assume that dimJ(FG1 F dimJ(FG) F
=
+
1 = t(G).
max {dimJ(FG)ei 1
Hence, by t h e foregoing,
G
has p-length
1.
'
z E 0
P'IP
(G)
GROUPS G
t(G)
WITH
=
U(p-1) + 1
427
We are now ready to prove the following result established by Motose (1980) and by Koshitani (1982a) for p # 2.
for p = 2
Let G be a p-solvable group with a metacyclic Sylow p-subgroup
3.3. THEOREM.
P of order pa, a > 1. t(G)
+
a(p-1)
=
1 if and only if P
If P
Proof.
If p = 2, assume that
is elementary abelian, then G
assume that t ( G 1 = a(p-11 is of p-length
+
Then
+
is of p-length
1 by Proposition 2.5.
1
Conversely
By Proposition 2.5, it suffices to verify that
1.
Let G
1.
$ Sq.
is elementary abelian.
(Proposition 3.16.20), so t ( G ) = a(p-1)
G
G/Op,(G)
be a counter-example of minimal order.
If
0 , ( G I # 1, then by Proposition 2.1,
P
a(p-1) and so t[G/O
P
,(GI
) =
hypotheses as G, case
1
=
t (GI 2 t (G/O
+
1.
a(p-1)
P
C/o ,(GI
b
10 (GI1 = p ,b 2 1.
P
If t ( O ( G I )
P
1
satisfies the same
P
Thus we must have
Thus t ( O ( G I )
P
=
0 , ( G I = 1, in which
P
3
b(p-1) + 1, then by
b(p-1) + 1, in which case 0 (GI
elementary abelian by Proposition 2.5.
P
> a [p-1) +
(GI )
and Proposition 2.1,
a contradiction.
0 (G)
P
But the group
a contradiction.
0 (GI # 1, say
Theorem 4.1.9
+
But P
is metacyclic, hence so is
p2.
and therefore 0 (G) is elementary abelian of order
by Lemma 1.3,
C
G
is
P
P (0 (GI) = 0 ( G ) .
Furthermore,
Thus G/O (GI is isomorphic to a subgroup P P P Because G is not of p-length 1, the group P is nonabelian.
of Aut(0 ( G 1 ) .
P
We now distinguish two cases. Case 1.
Assume that p = 2.
[C/O ( G ) l G 6.
P
Since G
By the foregoing, IPI 2 8, 10 ( G ) I
P
is not of p-length 1, we have
0 (GI = C (0 ( G ) ) is elementary abelian of order 4 P G P fully on 0 (GI by conjugation.
\ G I = 24,
and C/O (GI
P
4
and
in which case acts faith-
P
Let S = < a >
be a Sylow 3-subgroup of
G and 0 (G)
P
=
{l,z,y,Z!j}.
Since
CHAPTER 7
428
-1 (0 ( G I 1 = Op(G) I we may assme that a-'za = y and a ya = q. Consequently, G P N ( S ) n 0 ( G ) = 1 and (N(S1 1 = 6 by Sylow's theorem. We conclude therefore G P G that G is a semidirect product of 0 ( G ) and N(S) and so N(S) is the autoC
P G G Since the latter obviously implies that G z
0 (GI.
morphism group of
s , we
P derive a desired contradiction.
Case 2.
We now assume that p
If 0 CG1
is odd.
P
is cyclic of order p ,
G/O (GI is a p'-group, contrary to the assumption that G
is not of p-length
P
is elementary abelian of order p2.
Thus 0 (GI
P
IAut(0 (GI J
P
( P / o (G)
we have
P
I
=
1 or
I
= IGL(2,pJ
p.
.
length 1, a contradiction.
=
1.
Since
p(p-11 '(P+l),
is nonabelian and metacyclic, so P
But
and exponent p2.
of order p 3
1
then
It follows from Lemma 3.2 that G
and G / O p , ( G )
In view of Theorem 3 . 3 , the case where p = 2 separate treatment. For t h e r e s t of t h i s section
F
is
is of p -
S4
deserves a
is asswned t o be algebm-
i c a l l y closed. 3 . 4 . THEOREM. (Motose (19801).
e ,e 1
2
,. . .l e t
let Gi
Let p = 2, let G/O , ( G )
be all block idempotents of
be the inertia group of e
t(GJ =
I
FO
P
,( G ) .
P
IH$[
= 8
5
if
4
otherwise
Proposition 2 . 1 , that t ( G 1 Sylow 2-subgroups of Hil
>
4.
,(G1 .
and let
. ., t ) ,
Then
? - P
...,tI
for some i E t1,2,
Since the order of Sylow 2-subgroups of G
Proof.
S
For each i E {I,.
and put Hi = G . / O
i
"=
is 8, it follows from
zni
is the order of
...,t } .
Then each n
On the other hand, if
then by Corollary 2.16
n. Assume that and hence
kI
(H
= 24
lHil #
8
t ( G ) = 4.
and
Suppose that
for some k .
for all ak E Z2(Hk,F*). t(G) = t(S4).
t ( G ) G max I 2 KiGt lHil # 24 for all
Then Hk
2
'1
i
E {l,
[ H i [ # 8 for all i E {l,
S
and so, by Lemma 3.4.9,
...,t},
but
FakHk
Hence, by Proposition 2.15 and Corollary 2.16,
Thus t ( G ) = 4
by Proposition 3.1.
i
FS4
2
G WITH
GROUPS
1H.I = 8
F i n a l l y , assume t h a t
a. F 'H.
8
ai
t ( H . 1 = 5 by Theorem 4.3.4.
implies t h a t
Hence
t(G1 = 5
and by Lemma 3.4.9,
.
a s required.
429
i E {1,2 ,...,t 1 .
f o r some
d i h e d r a l group of o r d e r
E Z2(Hi,F*1.
+ 1
= U(p-1)
FHi
Hi
Then
is a
for a l l
By t h e foregoing, t h i s
W e now p r e s e n t two examples which i l l u s t r a t e t h e following f a c t s :
G
There e x i s t s a group
ti)
and a c e n t r a l p'-subgroup
fl such t h a t
t ( G ) # t(G/fl1 G
(ii) If
i s a s i n ( i 1 , then
t ( B ) # t (G)
where
B
Assume t h a t
p = 2
and l e t
is t h e p r i n c i p a l block of
FG . 3.5.
EXAMPLE. (Motose (19801).
group of o r d e r 3.
(121-'d(12)
that
G
Denote by =
d-l
and
t h e s e m i d i r e c t product of
A
by
.
Then
and l e t
a c t s t r i v i a l l y on
be a c y c l i c S4
such
t(G) = 4
by
Theorem 3.4. 3.6. EXAMPLE. (Motose ( 1 9 8 0 ) ) . c y c l i c groups of o r d e r
f
Then
3.
=
2
=
L e t a homomorphism
f
z2 x z2
x
and
S4
and
into
s3.
with r e s p e c t t o
S3+
:
can be regarded as a homomorphism of
a s e m i d i r e c t product of
M
p
Assume t h a t
GL(2,31
GL(2,31,
,
be
be defined by
since
s4
Let
G
f.
Then t h e following r e l a t i o n s
be t h e s e m i d i r e c t product of
hold:
-1 a ca = c Let
, a-lda
be a p r i m i t i v e 3rd-root of
E
e Then
=
e
cd
1 in
=
c
and
b-ldb =
6'
and put
= (1+Ec+E2c2)(l+d+d')
i s a block idempotent of
FM
a-lea # a Hence, i f
, b -1cb
such t h a t and
G ( e ) i s t h e i n e r t i a group of
Theorem 3.4 and Example 3.5, we d e r i v e
e,
b-leb = e then
is
I G ( e ) / M ( = 8.
Thus, by
CHAPTER 7
430
Furthermore, i f
t(BI
= t(Sbl.
3.7.
Remark.
B
FG,
is t h e p r i n c i p a l block of
then by Lemma 2.2,
t ( B 1 = 4.
Hence, by Proposition 3 . 1 ,
Similar examples can be constructed f o r an a r b i t r a r y prime
p
( s e e Motose ( 1 9 8 2 ) ) .
t(G1
4. COMPUTATION OF
Throughout t h i s s e c t i o n ,
WITH
M[p)
M(p)
denotes an e x t r a - s p e c i a l p-group of order
E Syl ( G )
P
odd, defined i n Sec. 2 of Chapter 4.
Note t h a t , by Corollary 4 . 2 . 4 , t ( M ( p ) ) =4p-3.
The information obtained w i l l be important f o r subsequent i n v e s t i -
subgroup.
Note t h a t i n case
i s of p-length
p = 3,
case
1
(Theorem 1 . 4 ) .
1.
with
M ( 3 ) E Sy13(Gl
(G)
E Syl
P
t ( G ) = 4p-3.
Hence, by P r o p o s i t i o n 2.3,
G
Mlp)
with
As w e s h a l l see below, t h e case where
c r u c i a l f o r t h e computation of from now on we assume t h a t
In
which a r e
F
t(GI
with
M(3) E Sy13 ( G I .
G
=
&d(3) is
For t h i s reason,
i s a f i e l d of c h a r a c t e r i s t i c
3
and begin by
G = Qd(3).
a t t a c k i n g t h e case where Recall t h a t t h e group
M(31
~ ( 3 = )
M = SL(2,31 a c t s n a t u r a l l y on t h e elementary a b e l i a n group
where
U is regarded as a vector space over t h e f i e l d F
group
Q d ( 3 ) i s defined t o be a semidirect product of
t h i s action.
U
3
of and
3 elements.
The
M with r e s p e c t t o
I t w i l l be convenient t o f i x t h e following n o t a t i o n :
H
= Qd(3)
a = Note t h a t
G
any p-solvable group
The most important example of such a group i s t h e group
(see Theorem 1.51.
The group
5
p
t h e r e e x i s t 3-solvable groups
not of 3-length Qd(3)
a s a Sylow p -
M(p)
Our aim i s t o extend t h i s r e s u l t t o p-solvable groups having
gations.
p3, p
[: i],
b =[!j
:]
and
c =
y]
in
M
=
SL(2,3)
COMPUTATION OF
t(G)
WITH
M(p) E Syl (G) P
431
Moreover, we put
where FT
T over F
is the group algebra of the group
(see Lemma 4.l(iv)
below]. 4.1. LEMMA.
With the notation above, the following properties hold:
f is a central idempotent of FM
(i)
(iil a 2 f = -f and fuf (iii)
'I
=
-f(u+u2) = fu2f
is central in FM.
(iv) ( ( a 0 b ) f ) '
=
0.
In particular, C
Hence
is a cyclic group of order
for all u E b'
3
T 2 = C2(l-Uob)f
consisting
of
and
commutes with T3 =
f
(a0b)f.
which implies that T
P-linearly independent elements.
-
(f-~)u(f-~) = ( ~ - T ) G ( ~ - T+) T+
(v)
(vi) Q+~Q+=
(u+-~)o+
for all u E U - 1
(vii) f(aobl3cQ' = G(y2-y)Q' (viii) (f-T)uM+ E C3M+ (ix) T+u&+(l-c) E C'M'
+ + TuM
(x] (xi)
+ + =MuT
calculation.
and
Q+x ( a 0 b ) f =
Q+(y-y2)z
and M+u(f-T) E M'C3 and
(l-c)Q'uT+ E M+C3
for all u E U for all u E U for all u E U
= O
+vQ T + ~ ( ~ - T ~ (i-c)
Proof.
for all u E U
= ( i - c ) ~ + v ( f - ~ ) u=~ o +
for all u , ~ E ) U
The validity of (il , (iil , (iv) and (v) follows by straightforward Note also that ( x ) is an easy consequence of (ix).
(iii) Invoking (i) and (ii), we derive
CHAPTER I
432
A s i m i l a r argument shows t h a t
(vi)
Because
Tb = T
and
're =
T.
Q i s both t r a n s i t i v e and r e g u l a r on U - 1 , Q+ u Q+
=
C
u4Q+ = ( U + - l ) Q +
&Q Applying ( i i ) ,w e have
(vii)
A s i m i l a r argument proves t h a t
(viii)
Invoking ( i l
, (iiil,
Q + ~ ( a o fb )= Q+ (y
- y2),
and ( v i i ) , w e derive
and a s i m i l a r argument shows t h a t
Because (ix)
Q is t r a n s i t i v e on
U-1,
t h e a s s e r t i o n follows.
Owing t o ( i ) ,( i i i ) ,and ( v i i l it follows t h a t
S i m i l a r l y w e have
we have
433
Since
Q is
U-1,
t r a n s i t i v e on
t h e a s s e r t i o n follows.
(xi) The proof of (ix) t o g e t h e r with (vii) and (x) ensures t h a t
A s i m i l a r argument shows t h a t
Because
Q
i s t r a n s i t i v e on
.
p r o j e c t i v e space follows.
U-1
and
M/ = PSL(2,3)
{ h , x 2 } , {y,y2}, { q , z 2 y 2 } , { s 2 y , q 2 2 }
,
i s t r a n s i t i v e on t h e the desired assertion
With the n o t a t i o n above,
4.2. LEMMA.
and
J(FH) = A + B + C Proof.
Note t h a t
4.l(i), ( i i i ) , FT
J(FS)Q+. t a t i o n s of F-dimension
Since
M, 10.
J(FS)Qi
is c e n t r a l i n {1,2,3}
where
is a n i l p o t e n t i d e a l of FM.
Thus
J(FM)
FM
and t h a t , by Lemma
c o n t a i n s both
J(FT1FM and
is t h e s e t of degrees of a l l i r r e d u c i b l e F-represen-
is a s p l i t t i n g f i e l d f o r
M,
the radical
J(FMI
On t h e o t h e r hand, one e a s i l y v e r i f i e s t h a t t h e s e t
is of
434
CHAPTER 7
consists of some F-linearly independent elements in J ( F T ) F M establishes the first assertion. natrual map FH--+ F(H/U) = FM
+
J(FS)Q+.
This
Taking into account that the kernel of the
is contained in J(FH) , we obtain J ( F H ) = A+B+C.
The next lemma is crucial.
4.3. LEMMA. (i) C'
=
The following properties hold: and C5 = 0
U'FM
and A 3 = 0
(ii) A 2
=
(iii) B'
5 T+FH + (J@-T) C' ( ~ - T I F H 5 T+C2(f-r)FH + (f-t)C2T+FH + T'C4
(iv) B 3 (v) B5
=
M'FV
0
(vi) BA'B
=
o
(vii) B 3 A
=
AB3 = 0
(viii) B*AB' Proof.
=
o
(i) Straightforward
/ii) Applying Lemma 4.l(vi), we derive
and
as asserted.
(iii) By Lemma 4.1(i) ,(iii), we derive
J@-T
is central in FM.
Applying Lemma 4.l(v),
COMPUTATION OF
tCGZ
WITH
M(pl
Syl
P
(GI
as required. (ivl
Invoking Lemma 4 . l ( v l t o g e t h e r with ( i )and (iii), we o b t a i n
a s desired. (vl
Applying ( i ), ( i i i l , ( i v )
and Lemma 4 . 1 1 ~ 1 , i t follows t h a t
a s required. (vil
By Lemma 4 . l ( v i i i l and ( i i ) ,we have
as a s s e r t e d .
435
CHAPTER 7
436
c - c3c2
+
0
+
c5
=o S i m i l a r l y w e have (viiil
Applying ( i i i l and Lema 4 . l ( i x l , (XI,we i n f e r t h a t
a s required. 4.4.
B 3 A = 0.
.
Proof.
H
If
LEMMA.
=
Qd(3I
and
p
then
= 3,
t ( H l = 9.
Applying Lemma 4.3, we s e e t h a t
(A+B15 c - FHmA3
+
FH*AB3
+
FH*B3A
+ B2AB2 + B 5 + FH*BA2B
=o Since
C5 = 0 ,
proving t h a t Put
x=
i t follows from Lemma 4 . 2 t h a t
t(Hl 1
9.
+ x + x2
. and
V =
together with Lemma 4 . 2 , we o b t a i n
1
+y +
y2.
Applying Lemma 4.1(1)
-
(iv)
437
and
whence
By Lemma 4.2, J ( F H ) *
contains XT'VT'.
Applying (11, we have
thus completing the proof. We have now come to the demonstration for which this section has been developed.
The following result is essentially due to Motose (1984a) who
established the crucial case where p = 3. 4.5. THEOREM.
Let G be a p-solvable group, p
P isomorphic to M(p1.
Then t ( G 1 = 4p- 3
odd, with a Sylow p-subgroup
438
CHAPTER 7
Proof.
G
Assume t h a t
t(G)
Proposition 2.3,
4p-3.
=
1.
is of p-length
1 . 4 ) , so we may assume t h a t
p 2 5
If
p = 3
“hen, by Corollary 4.2.4
G
then
G
and ‘chat
i s not of p-length
F
we may a l s o assume t h a t t h e f i e l d
Corollary 3.1.18,
t(G) # 9
A s s u m e by way of c o n t r a d i c t i o n t h a t
i s of p-length
and
1 (Theorem
By
1.
i s a l g e b r a i c a l l y closed. G
and l e t
be a counter-example of
minimal o r d e r . We set
W
0 ,(G), G
=
U
= G/W,
=
O (G )
1
3
3
1
and
P g
Let
g
and
= PWW / 1 PI G
G /U
=
2
1
1
g E G
be t h e images of an element
G
in
G2,
and
respec-
tively.
G
W e f i r s t observe t h a t
I u11 IP I
= 3,
33.
=
IG,I
then
IU
If
G2
IAut(U )
I
1 .
=
33,
By Theorem 1 . 6 ( i ) ,
xl , y l X
morphism
and
G
G+
e
.
PI
V
Q
5 Q.
. =
CG:X)
Then
IVI
If
Sylow 2-subgroup we must have
Q
1
1
G
Q
of
=
P1 , G
and so
If
contrary t o the f a c t t h a t
i s of 3-length 1, again a
3’
i s elementary a b e l i a n of o r d e r
U1
U
and so
, k 2 2,R 2 0.
,c y 1
1
1
=
1
t ( G ) = t(X1
SL(2,3).
.
<x , y > 1
under t h e n a t u r a l homo-
X and hence G 2
i s normal i n then
1
U 1 preserving t h e same r e l a t i o n s a s those
2
G =
<x
=
Hence we may assume t h a t
by Proposition 3.1.8
5 SL(2,3). V E Sy12(G2),
Hence, i f
has a normal Sylow 3-subgroup
lU1 I = 3‘.
If
SL(2,31 s i n c e
c
contrary t o t h e f a c t t h a t
Let
x
and
The l a t t e r implies t h a t
The group then
P
=
denote t h e inverse image of 2
(ill,
1
U 1 contains
We may r e p l a c e g e n e r a t o r s of
Let
U
then
by Lemma 1.3.
1
U1
and so
2
=
Aut(U )
Aut(u 1 = GL(2,3).
i s a subgroup of
between
I
Thus w e may assume t h a t
contradiction. and
i s contained i n
IVI > =
2
4,
[:
then and
V
5Q
=
P
,
contains the
.
Thus
= SL(2,3).
be a Sylow 2-subgroup of
GI such t h a t
U Q = 0 1 I
3 r 3
,(GI).
Because
CHARACTERIZATIONS OF GROUPS G
nU
WITH
1
it follows from Theorem 1.9 that G
439
9 U =1,
and Q
= 1
a- 1 t(G) = p +p- 1
1
U l and SL(2,3).
is a semidirect product of 1
Taking into account that SL(2,3) cAut(U 1 that G
I
IG
and any
c(
=
E
GI
is a quaternion group or a cyclic group and
23*33, it follows from Lemma 3.4.9 that for any subgroup H
FH
(2)
Let { e s ] be the set of all block idempotents of FW group of
e
Assume that W # 1.
in G. G
subgroups of =
and G/W
t(G/n = t(Is/W) < t ( G )
are isomorphic to M(3). then I s
of Gl
Z2(H,P*), EP"H
9
U 1 , we deduce
= Qd(3).
Since every 2-subgroup of since
acts naturally on
are isomorphic. for a l l
B
Then
and let I,
IG/Wl
q
IGI
be'the inertia
and the Sylow 3-
Hence, by assumption
Is/W
such that the Sylow 3-subgroups of
If the Sylow 3-subgroups of rs/V
are of order
Q
3*,
is of 3-length 1 [Proposition 3.16.20), so by Proposition 2.3 and t(Is/W) < 9.
Corollary 3.2.5,
Invoking (2) and Proposition 2.15, we therefore
conclude that
Thus W = 1 and
a contradiction.
G
= Qd(31,
contrary to Lemma 4.4.
This
completes the proof of the theorem.
5. CHARACTERIZATIONS OF GROUPS
G
WITH
t (GI
=
pa-l+p
-1
Throughout this section, G denotes a finite group and F an arbitrary field of characteristic p . order p a , (il
Recall that, by Theorem 4.3.4, if P
is a noncyclic group of
then the following conditions are equivalent:
t (PI = pa-'
+p - 1
(ii) t ( P I > pa-' (iiil P
contains an element of order p
a- 1
Our aim is to generalize the above fact to p-solvable groups. restrict ourselves to the case where p 2 3 .
In future we shall
The reason for this is as follows.
CHAPTER 7
440
p = 2
Let
.
= S
Then t h e Sylow 2-subgroups of
and hence s a t i s f y condition ( i i i l .
8
order
G
and l e t
G
a r e d i h e d r a l of
On t h e o t h e r hand, by Proposition
3.1.
t ( G 1 = 4 # 2'
p
Thus f o r
2,
=
+
2-1
Theorem 4.3.4 cannot be extended t o a r b i t r a r y p-solvable groups.
The following r e s u l t i s of independent i n t e r e s t and is t h e key t o t h e s o l u t i o n of our problem. 5.1.
THEOREM.
(Koshitani (1983),(1985a1, Motose (19851).
group with a Sylow p-subgroup
p = 2 Set
M
=
,
S
G/@
and l e t
t(G)
2 pa-1
0 , ( G I ,N = 0
P'PP
P
@/O ,CG)
and
@/oP , ( G )
be t h e F r a t t i n i
and t h a t
G
i s n o t of
is cyclic
P
4
(G), and
pr =
(N/M(.
Owing t o Corollary
F is a l g e b r a i c a l l y closed.
w e may harmlessly assume t h a t
3.1.18,
pa
be a p-solvable
Then
1.
Proof.
of o r d e r
Assume t h a t
0 (GI/Op,(G). P'nP
subgroup of p-length
P
G
Let
For t h e sake
of c l a r i t y , we d i v i d e t h e proof i n t o f o u r s t e p s . Here we prove that
Step 1.
that
is cyclic.
O/M
imply t h a t
G
Since
i s of p-length
G/N
is
isomorphic t o a subgroup of
(GL(2,pl 1 2.
=
p(p-11 (p2-11,
JlFGlP
and
the f i r s t assertion w i l l
B y Proposition 3.1.9,
a-r
GLf2,p)
we have
5 FG*J(FN)
and hence
oz Thus
If
t (N) > $-l
N/M
since
which c a s e
iV/O
is a subgroup of
t(N/M)
r-1
5F G ~ J C F N ) ~
and hence, by Proposition 2.3,
i s c y c l i c , then
G/N
a-1
JLWP
< pr
i s of order Aut(N/Ol
p
and hence
Thus
(Lemma 3 . 1 ) .
by Proposition 2.4.
< t(N/M) < pr
is of p-length N/M
1
i s noncyclic, i n
I t follows from (1) t h a t e x a c t l y
one of t h e following holds:
pP1
G
and
r
3
G
CHARACTERIZATIONS OF GROUPS
t ( G ) = pa-' + p
WITH
-1
441
or
N/M
N/@
I f ( 2 ) h o l d s , then by Theorems 4.3.4 and 1.10,
p2
order
i s cyclic.
@/M
and
GL(2,p)
Aut(N/@)
@/M
and
We now prove t h a t
Step 2.
p = 2
or
i s of o r d e r
p
x
~
o
f o r some
P
G/O
follows from Theorem 1.9 t h a t
p
is a subgroup of
T
=
e ,e ,...,e 1
2
of
e
in
i
T
where
be a l l block idempotents of
G,
1G
i
G
s.
FM
U. E
2
2.
GL(2,3)
and l e t
p
3.
It
and
G/N.
=
N/@
k
Gi
on
k
3
x
k
3
.
3 x k 3
Let
be t h e i n e r t i a group
Then, by P r o p o s i t i o n 2 . 1 5 , w e have
"i
H . = G./M, z z
Invoking
i s t h e s e m i d i r e c t product of
1
".
U.
G/N.
is a s e m i d i r e c t product of
t(G) = max { t ( F Hi) l G S where
~
so t h a t , by S t e p 2 ,
G L ( 2 , 3 ) with r e s p e c t t o t h e n a t u r a l a c t i o n of
and
1
p E {2,3).
p # 2
Assume by way of c o n t r a d i c t i o n t h a t
G/@
€
and i s conjugate t o
i s t h e minimal polynomial of every p-element o f
The aim of t h i s s t e p is t o prove that
Hence
i s a subgroup of
3.
=
G/N
Theorem 1.8, we t h e r e f o r e d e r i v e t h a t S t e p 3.
G/N
is cyclic.
[: y] (X-1)'
(3)
i s elementary a b e l i a n of
Thus i n any case
I n view of S t e p I, every p-element of
Thus
p2
is elementary a h e l i a n of o r d e r
t ( F 'Hi)
and
Z2(Hi,F*I
i s t h e nilpotency index of
J ( F 'Hi). Let
S be t h e s e m i d i r e c t product of
t h e n a t u r a l a c t i o n of homomorphism and l e t
[Hi:(Li
n Hi)] G 2 ,
SL(2,3)
L
i
on
k
3
x
k x k 3
a3 .
and
3
Let
be t h e i n v e r s e image of
SL(2,3) with r e s p e c t t o
f : G/M--t G/@
f(H.)
n S.
be t h e n a t u r a l
Since
it follows from Lemma 3.4.9 and Proposition 2 . 1 3 ( i i ) t h a t U.
= max { t ( L i n Hi)] lG<s 1GG (Li n H i ) / ( ( @ / M ) n Hi) i s isomorphic t o a subgroup of
t ( G ) = max
NOW
Ui
=
{ t ( F 'H.11
Theorem 4.5 and P r o p o s i t i o n 3.1.9, and Proposition 4.1.8, we d e r i v e
t(Ui)
< 9.
S and, by
Hence, applying Theorem 4.1.9
CHAPTER 7
442
= 3r
Completion of the proof.
Step 4.
@/M
By Step 1, t h e group group of G/O
< t(G)
3r = 3‘-’
contrary t o t h e f a c t t h a t
G/N
we have
GL(2,2l,
GL(2,2)
on
G/N
Since
i s a sub-
GL(2,Z). Applying Theorem 1 . 9 , w e deduce t h a t
=
Z2 x Z2 and
i s a semidirect product of
n a t u r a l a c t i o n of
s of 2-length 2 .
i s c y c l i c and G
Z
x
2
Z .
CL(2,2) with r e s p e c t t o t h e
Thus
2
G/Q
and t h e r e s u l t
S
follows.
p be an odd prime and l e t
Let
n 2 3.
Mn(p)
Recall t h a t t h e group
is
defined by
n-1
n-2
5.2. THEOREM. (Koshitani (1985al).
Let
p
P be a Sylow p-subgroup of
solvable group and l e t
G
be a p -
G
be an odd prime, l e t
p
of o r d e r
U
with
U
2 2.
or
Mu@)
Then t h e following c o n d i t i o n s a r e equivalent:
t (G)
(i)
=
pa-’
+
pa-’ G
(ivl
where
B
i s t h e p r i n c i p a l block of
FG
< t t G ) < pa
has p-length
1 and
P
i s isomorphic to e i t h e r
>3
for
a
(v)
t ( P 1 = pa-1
(vi)
-1
+ p - 1,
(ii) t ( B ) = pa-’
(iiil
p
4.
Z a-l P
x Zp
p- 1
p Q - l < t ( P 1 < pa
(vii) (viii)
P
i s noncyclic but has a c y c l i c subgroup of index
p
P i s isomorphic t o e i t h e r Z a-l x Zp o r Ma@) f o r u >
Proof. and 1.10.
Conditions (vl Hence, i f
G
-
( v i i i ) a r e equivalent by v i r t u e of Theorems 4.3.4
i s of p-length
equivalent by Proposition 2 . 3 .
3
P
1,
then conditions (i)
-
(viii) are
We a r e t h e r e f o r e l e f t t o v e r i f y t h a t each of t h e
c o n d i t i o n s ( i l , ( i i l , ( i i i l and ( v i i i ) imply t h a t
G
i s of p-length
Now C i ) obviously implies ( i i i l and C i i i l implies t h a t
G is
1.
o f p-length
1,
G
CHARACTERIZATIONS OF GROUPS
by v i r t u e of Theorem 5.1.
t(B)
Lemma 2 . 2 ,
i s of p-length
i s of p-length
G
which case
0 (G) # 1 since G
Suppose t h a t
Q
Let
G
Theorem 1 . 5 ( i I .
Q
z
z
$-2
P c 0 (GI. - P’tP
X
Y
and
X/Y.
x
a
i s of o r d e r
P,
Hence
G
2
Ma@)
i s of p-
G
then
a 2 3.
f o r some
To
0 ,(G) = 1 i n P
Y 4 X
Q
X/Y
and
i s metacyclic s i n c e
p3
and exponent
p,
p
P
5 0P ’ Q
CGl.
4 P.
Since
Clearly
$
Qd(p).
i s meta-
by v i r t u e of
Therefore, by Theorem 1 . 7 , i f
A
then
and hence
P
P
such t h a t
Then
This i s a c o n t r a d i c t i o n .
a n a b e l i a n normal subgroup of
zp’
x
we may harmlessly assume t h a t
has subgroups
On t h e o t h e r hand,
is of p-length
P
i s p-solvable.
be a Sylow p-subgroup of
cyclic.
z
2
Assume t h a t
1,
P
P
If
1 by Proposition 3.16.20.
prove t h a t
C/O , ( G I
and by
1.
F i n a l l y assume t h a t ( v i i i ) . holds. length
443
t ( B 1 > pa-’
Then
Hence, by Theorem 5.1,
P
G
1 and t h e r e f o r e
Assume t h a t ( i i l holds.
t(G/O r(G)l.
=
t ( G ) = pa-l + p - 1
WITH
A
is
E Z(P),
a P
and thus
1 as d e s i r e d .
i s of p-length
W e c l o s e t h i s s e c t i o n by i l l u s t r a t i n g t h e f a c t t h a t t h e converse of Theorem 5 . 1 does n o t hold.
Cil
G/Q
S
4
,
Namely, we g i v e examples of groups
@/02,(G1
(ii) G/Q 1 S4, @ / / 0 2 r ( G l
z2 2 z
t(G1 = 9 > 2
and
G
such t h a t
4-1
t ( G 1 = 7 4 24-1
and
The above examples w i i i i n d i c a t e t h e d i f f i c u l t y i n c h a r a c t e r i z i n g a l l 2-solvable
G
groups
with
t ( G ) > 2a-1
and t h e underlying f i e l d Q =
Let 0
and
T
F
where
2a
is t h e o r d e r of Sylow 2-subgroups of
i s of c h a r a c t e r i s t i c
< a , b l a ” = 1, b 2 = a 2 , bab-’
W
&.
2.
be t h e quaternion group and l e t
=
be automorphisms defined by
o(a) = b ,a@) = ab, ~ ( 0=)a-’, Then
G
U 3 = 1, -r2 =
1, TUT-’
= U2
be t h e s e m i d i r e c t product of
R =
and so
Q
R
and
~ t b =) a-’b
i s isomorphic t o
with r e s p e c t t o t h i s a c t i o n of
Then we have
02,(W)= 1 ,
o p
=
Q
i s t h e F r a t t i n i subgroup of
v/
2
s
s3
oz(w)
. R
Let
on
CHARACTER 7
444
For the r e s t of t h i s sectson, we assume that p
closed.
=
2
und F
is algebrae'cally
The latter condition does not impose any restriction, by virtue of
Corollary 3.1.18.
We need the following preliminary result.
5.3. LEMMA. CMotose (19851). Proof.
Put X
=
+
R 9FV
t(W) = 7.
With the notation above, and
Y
=
J(FQ)FW
.
Since J ( F Q ) F W C_ J ( F W ) ,
it
follows from Lemma 4.2 that J(FW1 = J(FR'R)FV + J ( F Q ) F W
= X + Y We shall show that X 3 = 0.
Since Y 5 J(FW)'
=
in C.
0 this will imply that
(X+Y17 = 0
+ + + -- 0 for + + + R xR yR = 0 for all
It suffices to verify that R CCR y R follow if we show that
=
all x,y E &, z,y E C = Q
Since {ll, { a 2 } and C are all orbits of &
and the latter will
- {l,a21.
Fix x , y
under the action of R,
we have
R + ~ R + ~ R=+ c+R+~R+= ( C + ~ ~ R + =
(a2+11 (a+b+ab)'R+
= o proving that t ( W I Since R+xR+
=
7.
C+R+
for x E C and a 2 + 1 and
Q
+
are central in F W ,
also have
This implies that t ( W ) > 6
5.4. EXAMPLE. (Motose (1985)1.
and hence that t ( W ) = 7 . Keeping the notation of Theorem 5.1, we shall
we
CHARACTERIZATIONS OF GROUPS G
+p - 1
t (G) = pa-'
WITH
2
z2
and
In what follows, we shall regard this as a homomorphsim from
w
to
present an example of a group G Consider the homomorphism R-
Let
M
=
G
G/Q
with GL(2,3)
,
IS
@/02,(GI
445
t ( G ) = 9 > 8.
defined by
GL(2,3).
denote the semidirect product of the elementary abelian group
x
Then clearly root of
M
of order
G/@
= 0 f(G),
and let e
1
3'
W with respect to the above homomorphism.
and
z
S b and @ / M S
be a block idempotent of
.
Let
FM
be a primitive 3rd
E
defined by
e = (1+Ee+~~e')(~+d+d') Then
# e
U-leu
and
-1 T eT = e
and so the inertia group
G(e)
of
e
in G
is
given by G(e) = < T , Q > * M
{el
Let
=, 'M
e
=
,...,e
e, e
let Hi = G(ei)/M, 1 =Si G 9. c1
i
E
Z2(Hi,F*1
1
be the set of all block idempotents of
Owing to Proposition 2.15,
a
t(F
and
there exists
such that
a. { t ( F 'H.)) z lGX9 a. is the nilpotency index of J ( F "Hi). t(G)
where
FM
iH.)
=
Since
a. t ( F 'Hi)
=
t(Hi)
by Lemma 3.4.9, we have
t(G1 If Hi 3.1.9.
max It(Hi)l lG59
contains no Sylow 2-subgroups of On the other hand, t ( H
contains an element T b
5.5.
=
EXAMPLE.
) =
of order
(Motose (1985)).
t(W1
G, = 7
= 7
< 8.
t(H.1
G 8
by Lemma 5.3.
by Proposition Since
8, we also have
Again, we preserve the previous notation.
aim is to present an example of a group G t(G1
then
with
G/@
S
,
Q/02 (GI
2
z2
Our and
CHAPTER 7
446
Consider the homomorphism
+ GL(2,31
and regard it as a homomorphism from W
defined by
Let G be the semidirect
to G L ( 2 , 3 ) .
product of the elementary abelian group M = < c > x < d > Then clearly M = 0 ,(GI, G / @
to this homomorphism.
,...,e
{e
= M+,e
or
*M
S
and W and @/M
be the set of all block idempotents of FM.
}
with respect 2
z .
Let
Then G(ei) = G
and so
by virtue of Proposition 2.15 and Lemma 3.4.9.
by Proposition 3.1.8(ii)
Now
and Corollary 4.2.4(ii).
t(W1
Thus
= 7
by Lemma 5.3 and
t(G)
=
t ( W 1 7 7, as
required.
“hroughout this section, G characteristic p .
is a finite group and
F is an arbitrary field of
We know, from Theorem 4.3.6, that if
P is a group of order
pa which is not elementary abelian, then the following conditions are equivalent: (il
t ( P ) = ( a + l )(p-1) + 1
(iil t ; P ) < (a+2)(p-1) + 1 (iii) There exists a central element z
of order p
in P
such that P / < z >
is
elementary abelian. Our aim is to extend this result to p-solvable groups under certain conditions. Recall that the prime p
is said to be a Fermat prime if p = 2n + 1 for some
n a 1 (the latter, of course, implies that n is a power of 2). 6.1. LEMMA. (Motose (1984d)). group P
of order p
regular.
,a
2 1.
If t ( G ) < (a+2) (p-1)
2-subgroups of Proof.
a
C/O , ( G I
P
Let G be a p-solvable group with a Sylow p-subAssume that G is of p-length
+
1, then P
a 2 and P is
is a Fermat prime and the Sylow
are nonabelian.
Owing to Proposition 3.16.20,
.?
is nonabelian.
Hence, by Theorem
p
l.ll(iii), assume t h a t
i s odd.
,
0 ( G ) = 1. P
U
0 (GI, @ P
=
w e may harmlessly
(Lemma 2.21,
P
Put
IUI = PSI S 2 1 .
write
t ( G ) 3 t(G/O ,(GI)
Since
U and
t h e F r a t t i n i subgroup of
By Theorem 4.1.9, w e have
t(G)
z t ( U ) + t(G/U) -
1
while by Proposition 2.1,
Hence w e deduce t h a t
I t follows, from Theorem 4.3.6,
<x,U>'
r e g u l a r and
5U
for a l l
(XU)P =
u
that
X E
I s o f exponent
P,
p.
Because
P
is
it follows t h a t
2
for a l l
x
E
P,u E U
Hence we have
..+x+l = u2-I ... uxu
;Cp-l+. Iorall
X E P
and
...,x,l
sum of endomorphisms
GL(U/@).
Sylow p-subgroup
P
of order
Sylow 2-subgroups of ( i ) t ( G ) = a(p-1)
P
+ 1 or
+
i s of p-length
P/
Let
G
>
1.
pa,a
G/O ,(GI
(ii) t ( G ) < (a+2) (p-1)
that
U.
of
x-p(,lp and
= 1
(1)
$-l+...+x+l
By Lemma 1.3,
G/U
isthe
.
i s a subgroup of
Invoking (1) t o g e t h e r with Theorem 1.8, t h e r e s u l t follows.
6.2. THEOREM. (Motose (1984811.
(iii) G
X -s s u s = x ux
where
U E U ,
=
be a p-solvable group with a r e y l a r If
p
i s n o t a Fermat prime o r t h e
a r e a b e l i a n , then t h e following are e q u i v a l e n t :
t(G) =
(a+l)(p-1) +
1
1 1 and
P
has a c e n t r a l element
z
of o r d e r
p
such
i s elementary a b e l i a n
In p a r t i c u l a r , by Proposition 2.5,
t ( G ) = a(p-1)
+
1 i f and only i f
P
is
elementary a b e l i a n . Proof.
By Lemma 6.1,
our assumption on
G
forces
G
t o be of p-length
1.
CHAPTER 7
448
Hence, by Proposition 2.3,
t ( G ) # a(p-1)
+
t(G)
If P
t(P1.
=
1 by Proposition 2.5.
Hence in this case (i), (ii) and (iii)
In case P
are equivalent by Theorem 4.3.6.
not elementary abelian, then
is
is elementary abelian, then (i),
(ii) and (iii) are always true, again by Proposition 2.5.
This completes the
proof of the theorem.
Let G
be a finite group and F
ODD.
an arbitrary field of characteristic p . U
, a?
is a group of order p
to Theorem 4.3.5, if P
p
t ( G ) =pa-’,
7. CHARACl‘ERIZATIONS OF GROUPS G WITH
Owing
1, then the following
conditions are equivalent:
t ( P ) = pa-’
(i)
(ii) P = M ( 3 )
or
P = Z xjz X Z 2
2
2
Our aim in this section is to extend this result to p-solvable groups where p The case p
odd.
= 2
seems formidable for the following reason.
is
Assume that
p = 2 and consider the following three groups: G G
1
2
G3
= Z x z 2
= S
4
x Z
x Z
2
2
x Z
3
x Z x Z 3
3
xz!
3
3
is the group constructed in Example 3.6.
These three groups have the same order
2333.
Furthermore, by Corollary 3.2.5,
Proposition 3.1 and Example 3.6, we have
t(Gl) = 4
t(G3) = 5
t(G ) = 4
The above shows that there exist two groupsI namely
GI
and
GpI
with the same
nilpotency index whose Sylow 2-subgroups have different structures. hand, there are two groupsI namely
G2
and
GB,
On the other
with different nilpotency indices
whose Sylow 2-subgroups have identical structure.
This explains why we shall
restrict ourselves to the case p 2 3.
7.1. LEMMA. Then
t(G) =
Proof.
Let G
pa-1
be a p-solvable group of p-length
if and only if P
M(3)
or
Direct consequence of Theorem 4.3.5
P
z
U
1 and of order p ,a x Z2 x Z2
.
and Proposition 2.3.
1.
CHARACTERIZATIONS OF GROUPS G
Let G
7 . 2 . THEOREM. (Motose (1986all.
of order pala > 1,p odd.
subgroup P and P
2
If p
=
and P
3
IGI.
argue by induction on
2
has p-length at least 2 .
Then t ( G ) = pa-'
if and only if p = 3
=
Furthermore, by Corollary 3 . 1 . 1 8 ,
N/@
group of G L ( N / @ ) , where
N/@
2
M ( 3 ) , we
Then G/N
i s
(GI
, pr
is not cyclic. G/N
and hence
is of p-length
I N/MI
and
is isomorphic to a sub-
is regarded as a vector space over
cyclic of order p
=
P (Lemma 1 . 3 ) . P Indeed, if
is a p'-group,
2.
G L ( 2 , 3 ) acts naturally on the elementary abelian group E
of
Let T
order 9.
and P
P'tP
P
contrary to the assumption that G Every subgroup
by Theorem 4.5.
we may also assume
Put M = 0 ,(GI I N = 0
has p-length at least two, the group N/M
i s cyclic, then
9 = 33-1
To prove that p = 3
by the Frattini subgroup of N / M .
Since G
t(G)
Owing to Lemma 7.1, we may harmlessly assume that G
i s algebraically closed.
let @/M
GL(2.3)
be a p-solvable group with a Sylow p-
M ( 3 ) , then
t ( G ) = pa-'.
Conversely, assume that
N/M
449
M(3).
Proof.
that F
t ( G ) = pa-1, p ODD
WITH
(respectively, Sl
(respectively, S L ( 2 , 3 1 )
be the semidirect product of E
with respect to this action.
of
and
We divide the
rest of the proof into a number of steps.
P I
Step I .
of
Here we prove that if
GL(2,31
(hence a
=
J ~ F N P#
r + l l and
r-1 Assume that J ( F N l p # 0.
]@/MI
=
0, 3r-2
then p
=
3,
C/N
is a subgroup
.
Then, by Proposition 2.3,
It follows, from Theorem 5.2, that
r- 1 + P - l
t(N1 = p and that N / M
I
z
r-l x
z
P
P
or N / M
minimal number of generators of N / M
2:
Mr(p)
for 1 2 3 .
is two, we conclude, from Lemma 4 . 3 . 1 ,
N/@
is an elementary abelian group of order p'-
Now
G/N
GL(2,p)
is a subgroup of
Since in both cases the
In particular,
I @/MI
=
that
pr-2
GL(N/@) = G L ( 2 , p ) , and every element of order p
in
is conjugate to
[A' ?]
for some 0 # A e P p
CHAPTER 7
450
(X-1
Thus
is the minimal polynomial of every element of order p
)*
Since p
is odd, Theorem 1.8 implies that p = 3 .
Step 2.
Here we prove that
lQ/MI
8-2 and
=
p = 3,
+
t ( N I G 3p1
r- 1
i s a subgroup o f
GL(2,3) (hence a = r + l ) ,
2.
Then t ( N ) G pr-'
= 0.
Assume that J ( F N I p
G/N
in G / N .
and, by Proposition 3 . 1 . 9 ,
a-r S(FGIp
But, by hypothesis, t ( G )
-
=
pa-'
FG*J(FN)
and so, raising both sides to the power of
1 yields
Thus we must have
Since p
I Q/Ml=
implies that N / M
is odd, Theorem 4.3.5 N/Q
3r-2,
is elementary abelian of order
M(3). 3'
Hence P G/N
and
=
p
=
3,
is a subgroup of
GL(2,3).
Invoking Step 1, we deduce that p
IQ/MI
= 3r-2
Step 3.
.
G/N
G/@
is a subgroup of
3'-l
+
is a semidirect product of N / @
and
G/N
T.
Let f
:
G/M+
G/Q
be the natural homomorphism, let L
4.1.9
Since U
Then
2
>
(H:V).
together with Proposition 3 . 1 . 8 ( i i ) , we derive
Set U = V/((F/M) n H )
< 9.
and
,
3r = 3a-1 = t ( G ) = t ( f H I
t(U)
2.
By Proposition 2.15, there exists a block idem-
be the inverse image of f ( H ) n S and let V = L n H . Invoking Lemma 3.4.9
and
G/@ = S.
of FM such that for H = G(e)/M
for some ci E Z2(H,F*).
GL(2,3)
is a subgroup of t(N)
Reduction t o the case where
hence G / @
3,
Furthermore, by (1) and ( 2 1 , we also have
By Step 2 and Theorem 1 . 9 ,
potent e
=
=
t(V)
and observe that, by Theorem 4 . 5 and Proposition 2 . 1 ,
is isomorphic to a subgroup of S ,
and Proposition 4.1.8
that
it follows from Theorem
CHARACPERIZATIONS OF GROUPS G
WITH
where the latter equality follows from Step 2. Since
t ( U ) = 9 and M(3J
2.1 and 2.4 that U
=
group of
S.
of order 4.
Step 4 . Let f
, p ODD
>_ @/M
Thus H
t ( U ) = 9.
v#
Hence a 3-Sylow subgroup
Then, by induction, t ( V )
G/M.
Thus we may assume that N
has 3-length at least 2, so G/N
=
T
3
is a 3-
Thus G / N = SL(2,3J since G/N
must contain an element
contains an element of order
3.
as required.
Completion of th e proof. :
FG+
F(G/@)
=
FS
be the natural homomorphism.
Then, by Lemma 4.2 and
the proof of Lemma 4.4,
where
and
451
U = G I @ is a subgroup of S which contains a Sylow 3-sub-
But G
Hence G/@ = S ,
a-1
and so D is isomorphic to P.
3ai1 = '3
is isomorphic to M(3).
and
p
is of exponent 3, it follows from Propositions
Assume that either M # 1 or
group, V = G
=
contains a Sylow 3-subgroup of S.
D of V is of order 271@/MI
implies that D
t(G)
Z
such that Z 5 = 0.
is a right ideal of FS
Since
it follows that
JWG) = f-l(Z1 + J ( F N ) F G On the other hand, it is easy to see that
(f-'(Z)
1
5 Kerf
= J(F0)FG
and so
Taking into account that
t(N)
+ 2
(Step 2 1 , we derive
CHAPTER 7
452
Thus F = 2
Theorem l . S ( i ) ,
.
@ = 1, which by S t e p 3 implies t h a t
and
P
M ( 3 ) and t h e r e s u l t follows.
G = S.
Hence, by
453
8 Radicals of blocks Let
B
be a block of a group a l g e b r a
FG
d.
with d e f e c t
d
motivates t h i s chapter i s t o discover r e l a t i o n s between
t ( B ) of
index
J(B).
on lower bound f o r
The problem t h a t and t h e nilpotency
The f i r s t two s e c t i o n s provide some g e n e r a l information
t(B)
and upper bound f o r
t(Z(8)). The r e s t i s based on t h e
Fong correspondence and t h e Khshammer' s s t r u c t u r e theorem f o r blocks of p s o l v a b l e groups.
The l a s t s e c t i o n provides a number of important a p p l i c a t i o n s .
G
Among o t h e r r e s u l t s , w e prove t h a t i f and
t ( B ) = pd
i f and only i f
1. A LOWER BOUND FOR
t( B )
istic
p > 0.
denotes a f i n i t e group and
Given a block
index of t h e Jacobson r a d i c a l As u s u a l , w e w r i t e
t h e exponent of integers.
A,
G,
CR(G1
B
J(B)
B
of
and exp(G)
Given a s u b s e t
F a f i e l d of character-
6(B)
and
G, n E
x+=
=
Cx fix
alaPn E (if
n
xp
1.1. LEMMA.
Let
=
N
and a finite-dimensional F-algebra
n E No and l e t A
f o r some
m2 I
tA,AI}
X = n
rP
a,
IXE
then by d e f i n i t i o n 'X
be a finite-dimensional F-algebra. and
T ( A ) = u Ti(A) i=0
(ii) There e x i s t s
rn E No such t h a t
= 0)
XI
a0
( i ) [ A , A ] 5 T n ( A ) , !?',(A) 5 T n + l ( A )
and
denote t h e s e t of a l l nonnegative
~ 1 2E I A , A I I
{a E
G
No
m
TJA)
B.
f o r t h e d e f e c t group of
we w r i t e
T ( A ) = {a E
d
6( B ) .
f o r t h e s e t of conjugacy c l a s s e s of Let
X of
d(p-1) + 1 Q t (B)< p
FG, we w r i t e t ( B ) f o r t h e nilpotency
of
respectively.
then
is cyclic.
6(Bl
I N TERMS OF THE EXPONENT O F
G
Throughout t h i s s e c t i o n ,
i s p-solvable,
Trn(A) = T ( A ) .
454
CHAPTER 8
Proof. (i) Direct consequence of the property
x
[A,AI
E
P
implies
E [A,AI
established in the proof of Lemma 2.2.1.
...,aS
(ii) Choose a basis u l , u 2 ,
iE
for the F-space T ( A ) .
there exists m : 2 1 such that u?
{1,2,. ..,s}, S
2 =
Z Aiui with hi
F, we have for rn
E
i=1
rn
s C
2
5 ",(A)
For any p-subgrour, D C E
Hence, given
0 (mod I A , A l )
and hence that T,(A)
of
5 D.
6(C)
Ck(G) with
0 ,
and l e t
+
be t h e exponent
.
1.
pn be t h e exponent of t h e
i s a b e l l a n and l e t
n =
o
for a l l
z E JCFGI.
Apply Theorem 1 . 4 t i l ) .
1.7. COROLLARY.
If
t
is t h e exponent of Sylow p-subgroups of
zt = Proof.
FG
D i r e c t consequence of Theorem 1 . 4 ( f ) .
1.6. COROLLARY.
Proof.
B
Let
o
G,
then
z E J ( Z (FG)
for a l l
Apply Theorem 1 . 4 ( i i ) .
We next i l l u s t r a t e t h a t Theorem 1.4U.i) need not be t r u e f o r
z E J(B).
The
following two observations w i l l c l e a r our path.
1.8. LEMMA.
(i)
Let
(R1 x R 2 x
modules.
R,RI,R2,...,R n be r i n g s R G x R G x x Rn ) G
...
and l e t
... x RnG
G,GI,G2
be groups.
a s r i n g s and
R
x
... x Rn-
CHAPTER 8
458
(ii) R(G x G 2 ) 1 (RGIlG2 as rings and R-modules (iii) For any
k
>
1, M (RIG 4 M (RG) as rings and R-modules k k
(i) Put S
Proof.
projection map.
=
R
1
Denote by
(7
Kerfi = 0
:
i
... x Rn
x
2
71
n Then
x R
SG-
and let fi
RiG
Ri be the
: S+
fi'
the ring homomorphism induced by
i
and KerTIi = (Kerfi)G, 1
n.-
Consider the map
i=1
1
... x RnG (z) ,... (33 1 -.
R G x R2G x
SG-
z C-L
(z) ,TI
(71
l i
,TIn
2
is a homomorphism of rings and 5'-modules.
It is clear that IT
n n n KerTI = n Kerni = n (Kerfi)G = t n Kerfi)G
i= 1
i=l TI
is injective.
To prove that
,...,ynl
and write
yi = Cr .g gs
with r
gi
be such that f . ( r i = r 1c 2 g gi'
E
... x
E R G x R G x
Ri, 1 c i
$G n .
TGr
gl
9
0,
i=1
is surjective, fix
TI
(yl,y2
=
Since
For any g E G,
n.
Then
RnG
Ti(Crggl =
let P
9
E S
yi, 1 G i G n , and thus
,... , ~ , l ,
(Y,,Y,
=
as required. (ii) By looking at the chain of rings R
-
is a free R-module with
(g1,g21
{ g l g 2 1 g lE G 1 ,g 2
5 RGl 5 ( R G l I G 2 , we E
G21
as a basis.
infer that
(RG1)G2
Thus the map
g l g 2 induces an isomorphism R ( G l x G2)-
of R-modules.
(RGl)G2
Since this induced map obviously preserves multiplication, (ii) is
established. (iii) Let e . . be the n x n 23
elsewhere, and let ei zero entry elsewhere.
J,g
matrix with (i,jl-th entry
be the n x n
M k I R ) G - + Mk(RG) of R-modules. preserves multiplication.
A
and
induces an isomorphism
routine calculation shows that this map also
So the lemma is true.
1.9. LEMMA. Let G = H x P , where k FH (Oil for suitable k,ni i=1 "i
nM
matrix with (i,j)-th entry g E G
e..gW e 23 i , j ,g
Then the map
1 and zero entry
P
is a p-group and H is a p'-group.
and division rings Di.
Then
Write
459
M,
(ii) Each
(Dip) is indecomposable
i (iii)
P i s t h e d e f e c t group of any block of
(i) D i r e c t consequence of Lemma 1.8.
Proof.
D.P
(ii) I t s u f f i c e s t o show t h a t each
D.P.
zero c e n t r a l idempotent of
p.
characteristic
e
(ii) L e t C
C_
Suppe.
i s indecomposable.
e E Z(Di)P
Then
Hence, by Theorem 2.2.4,
C
By Theorem 2.2.4, of
5H
e,
e
FG
be any block idernpotent of
P i s t h e d e f e c t group
Thus
FG.
=
as required.
be a non-
a s required.
C E Ck(G1
and l e t
and hence
e
Z(Di) i s a f i e l d of
and
1
Let
.
P is
be such t h a t C.
t h e d e f e c t group of
I t i s now an easy m a t t e r t o show t h a t Theorem 1.4(111, need not be t r u e f o r
z E J(B). (Kilshammer (19821 I .
1.10. EXAMPLE.
P
=
X
verify t h a t
F ( H x PI
FH
21
be an elementary a b e l i a n group of o r d e r
4.
let
Bo
has a block
has a block B
J(FP) =
Then
be an a l g e b r a i c a l l y c l o s e d f i e l d of
H be a nonabelian group of o r d e r
2,
characteristic
F
Let
I(P)
1
M (Fp). 3
and so
J(B1
M3(F1. Let
and l e t I t i s easy t o
Hence, by Lemma 1.9, t h e group algebra
FP.
T ( P ) be t h e augmentation i d e a l of
M3(I(P)).
Identifying
J(B)
and
M3(I(P)),
we see t h a t
g-1
0
h-1
0
0
0
0
0
0 z =
but
' 2
# 0.
.
2 . AN UPPER BOUND FOR
t(Z(B1)
Throughout t h i s s e c t i o n , istic of
2.1.
p.
denotes a f i n i t e group and
F
a f i e l d of character-
Our aim i s t o provide an upper bound f o r t h e nilpotency index
J ( Z ( B ) ) , where
LEMMA.
G
Let
B
P
i s a block of
t(Z(B))
FG.
be a p-subgroup of
G
and l e t
H be
a subgroup of
C
such
CHAPTER 8
460
that
D,
C(P) G let p
5 H C_ NIP). :
e
where each
Z(FH)
PG wfth d e f e c t group
be t h e Brauer homomorphism, and l e t p ( e ) = e
FH.
i i s a block idempotent of
Z G Proof.
C ,C
Let
I (G) D
let
e be a block idempotent of
Let
Di
If
+ . .. + e n' ei f
is a d e f e c t group of
5 D.
D.
then
G Z(FG1-
1
,...,Ct
be t h e F-linear
Proposition 2 . 2 . 6 ,
L E Ck(H)
L
and
x E G.,
f o r some
2
that
5 Ci
+ + C1 , C p , .
span of
IDCG)
i
6 ( C i ) C D and G We know, from Lemma 2.2.5 and
,Ci.
Z(FG1
e E ID(G1 I
e.
containing
If
w e conolude t h a t
C I XEG
I (HI DxnH
with
then
E {1,21.,.,t},
P(e) E
Takinq i n t o account t h a t
..
i s an i d e a l of
f o r some
Because
CR(G)
be a l l elements i n
(HI D"~H Z(FH1
is an i d e a l of
ei = e i p ( e ) ,
and t h a t
we see t h a t
e E Hence, by Lemma 3.16.5,
D . C Dx
%a
2.2.
C1
5D,
nH
as required.
G,
of
and t h e r e f o r e
H
let
El.
be a l l d i s t i n c t p-regular c l a s s e s of
Zi
(i)
c1,c2,. ..,zr
be t h e image of
iE
=
PC(P1,
and l e t
For a l l
Proof. As
i
For each
Ci under t h e n a t u r a l homomorphism G -
a r e a l l d i s t i n c t p-regular c l a s s e s of
,...,rl. c E {c ,...,cr}, C 5 C(P1 G C E { C ,-..,C 1, 6 ( C l / P r
H/P
.. , r } ,
E {1,2,.
G/P.
and
I cil
Then =
lcil
{1,2
(ii) For a l l
E C.
G n)
9
P be a normal p-subgroup
Let
let
x
x E G
f o r some
G
,...,C
(iiil
i E I DxnH (HI
i
(1 Q
G
LEMMA.
for a l l
e
C I CHI x€GD%H
Let
x
IGI = pnq, i s a p'-element
where
and
P
6(C)
i s a d e f e c t group of
( p , q ) = 1.
and as
5
Fix
C E {C
-
C.
,...,C
}
and l e t
461
g E G,
For any
let
cii(a1
Let
By (ii), we have
H/P.
=
where
h-lxh
Then
E Cz(Z1.
k,
element, say of o r d e r
=
and
d = 1.
P
We next show t h a t
CHCS)/P
=
5 CH(xl
d3c
(1)
d
is a p-element, it follows t h a t
d E P.
f o r some
(h-lxh)k = xkdk
x is
Since
a p'-
dk
=
and (11 i s e s t a b l i s h e d .
H (2)
5 C*(Z).
C,(rl/P
and t h e r e f o r e
=
h E C
Thus
G/P.
in
xd
1=
and hence t h a t
g
be t h e image of
Since (1)
implies ( i i i ) ,we a r e l e f t t o v e r i f y ( i ) .
ICI
To t h i s end, we f i r s t observe t h a t
(E:Cz(G) 1 X ,X
Assume t h a t
Then
-1
yx,y
x1
=
3:
1
d
and
.
yx2y
E
H
{c ,...,cr}
f o r some
commute.
X
Thus
Y
Finally, l e t
y
E
yx y-ld
-1
=
2
y,
o r d e r of
E Y.
y m E P.
then
+
ipn
such t h a t
X2,
and
z
d E P.
jm = 1.
Then we
ma71
C E {Cl
Let
y
xi
E
56(x2)
X i =1,2. i'
and so
we have
are distinct. of
H/p
C(P1.
and l e t
If G there are integers
Because (p,m) = 1,
E
m
i
is the
j
and
Hence
i s t h e h i g h e s t power of
element.
,...,Cr-
assume t h a t
. n .
pn
P
a r e p'-elements,
proving t h a t t h e
y As
and l e t
By ( i t ) ,
yx y - l
and
z2,
=
denote a conjugacy c l a s s of p'-elements
be such t h a t
-
=
(H:CH(~))
a r e such t h a t
y E H
Since
=
s i n c e by (1)
=
,...,Cr 1
=
p
.n
p yJm E p p
(2)
G , yipn
d i v i d i n g t h e o r d e r of
is a
p'-
.n be such t h a t
yzp
c.
E
Then, by ( 2 1 ,
Y
=
?
and t h e r e s u l t follows.
2 . 3 . LEMMA.
Let
P be a normal p-subgroup of
G,
H
let
=
PC(P1
and l e t
G
n
:
of
FG+ FH
F(G/P1 be t h e n a t u r a l homomorphism. Di be a d e f e c t group of
and l e t
For each proof.
bi, 1 G i
,...,~
( 1b a r e a l l d i s t i n c t blocks of
iE
,...,n},D./P
(i) n ( b l ) , n ( b 2 )
(iil
Let
(i) L e t
{1,2
b ,b 1
ei
,...,bz
be a l l blocks
n.
F(H/P)
i s a d e f e c t group of
4 = n(Z(FHFH)) and l e t
2
n(bi).
be t h e block idempotent of
FH
462
CHAPTER 8
bi, 1 4 i
contained i n
4
n.
By Proposition 3.1.1,
and t h e r e f o r e , by Lemma 3.4.16, idempotents of
T.
idempotents of
F (H/P).
of
ci
C',Cr, 1
2
1
...,T ( en
2
be a l l p-regular c l a s s e s of
a r e a l l d i s t i n c t p-regular c l a s s e s of
(iil
e
Let
FH
be a block idempotent of
r e g u l a r c l a s s e s of
H
such t h a t
Suppe =
and l e t
c
suppn(e) = C r U where
Cl denotes t h e image
H/P.
Hence, by Theorem
1
,.. U Ct.
U
be a l l p -
C ,C ,,..,Ct 2
Then
... u c; 4 G/P.
Ci under t h e n a t u r a l homomorphism G
i s t h e image of
CI
If
F(H/P).
T contains a l l c e n t r a l kdempotents of
2.2.4,
H.
T contains a l l c e n t r a l
G/P, then by Lemma 2 . 2 ( i ) ,
under t h e n a t u r a l homomorphism G-+
...,C f
are a l l d i s t i n c t primitive
I t t h e r e f o r e s u f f i c e s t o prove t h a t
C ,C f...,Cp
Let
lT(e 1 ,
i s a nilpotent ideal
KerT
The
d e s i r e d conclusion now follows from Lemma 2 . 2 . ( i i i l . 2.4. THEOREM. (Okuyama (1981)).
Let
B
=
B(e)
be a block of
FG
with d e f e c t
d
Then
proof.
=
Z(FG)e,
k k 2 1, J ( Z ( B ) ) ' = J ( Z ( F G ) ) e .
all
If
Z(B)
We have
d = 0
is true for write
D
J(B)
then
d
=
0
=
so
J ( Z ( B 1 ) = J ( Z ( F G ) ) e and t h e r e f o r e f o r
Thus we need only v e r i f y t h a t
0 (Theorem 3.6.4)
and t h e r e f o r e
and w e argue by induction on
f o r a d e f e c t group of
B.
d > 0.
S ( Z ( F G ) ) e = 0.
Thus ( 3 )
I n what follows, w e
For t h e sake of c l a r i t y , t h e rest of t h e
proof i s divided i n t o t h r e e s t e p s .
S t e p 1. H = C(P). G
Let Let
g E D be an element of o r d e r p n , n > 0,
let
P
p : Z(FG) -+ Z(FH) be t h e Brauer homomorphism.
=
and l e t
I t w i l l be
shown t h a t
By hypothesis, t h e r e e x i s t s
C E C&(G)
with
C 5 Suppe and 6 (C) = D.
Hence
463
H
0
C # @ and so, by Theorem 2.2.3, p(e1
=
sum
natural homomorphism. F(H/P)
Ker'll = FH(g-11,
is at most d .
Let
n(f)
Then, by Lemma 2.3,
with defect at most
d-n,
+ em
Fix f E { e
of block idempotents of FH.
Lemma 2.1, the defect of f
Therefore we may write
.-.
+ e2 +
e 1
as a
# 0.
p(e1
,...,e 1
TI
and observe that, by
m
: FH+
F(H/P)
be the
is a block idempotent of
since the defect of f is at most d.
Since
it follows from (3) by induction hypothesis that
d-n
f 5 FH(g-1)
J ( Z ( F H 1 IP
Since pd - 1 2 pd-n (pn
- 11,
(5)
raising both sides of (5) to the power of pn
-1
yields
i=o
where the last equality follows by virtue of Lemma 3.11.2.
proving (4) by virtue of Step 2. Zp,
=
,... ,S r
Sl,SZ
Let 21
I: FS: i=1
p(J(Z(FG1)
.
5J ( Z ( F H 1 ) .
be all p-regular sections of
d
be a p-regular element of G
such that 3cg = gx. the coefficients of ficients of
z
G and let
We next prove that J(z(FG)~P -le
Let x
Thus we have
and
g
c
z
(6)
- Pr
It suffices to show that, for any z 3c
and leg
in
Z
are equal.
are constant on conjugacy classes of
G
This is so since the coef-
G.
We may harmlessly assume that either x E Suppz or xg E Suppz. a
defect group of e .
Then, by Lemma 2.2.5
n
of order p d in J(Z(FG))' -'e,
a p-element of
and Proposition 2.2.6,
Let D be
ID(G) is an
CHAPTER 8
464
ideal of
Z(FG) containing e .
x E Suppz or xg
group of
D.
Since
Thus we may assume that g belongs to a defect
,H
Let P = < g >
e.
z E T,(G).
it follows from the definition of ID(G) that g
E Suppz,
belongs to a conjugate of
In particular, we must have
C(P) and
=
P :
Z(FG)
Z(FH)
f
the Brauer
G
homomorphism.
Since x,xg E H,
same as the coefficients of x
the coefficients of x
and zg P(Z1
in P(z).
and xg
in z
are the
But, by (4),
pn-li E FEI( c g 1 i=o
and so the coefficients of x
Step 3.
and xg
Completion of the proof.
in
are equal, proving ( 6 )
p(Z)
Owing to Theorem 3.3.14, we have
J(Z(FG))Zp, = 0 Hence, multiplying both sides of (61 by
J(Z(FG)) we obtain d
cPCZ(FG)lP e
= 0
proving ( 3 ) and hence the result. 2.5. COROLLARY.
G
Let
a
be a group of order p m, ( p , m ) = 1.
Then
a J(Z(FG1Ip Proof.
Let e ,e r . . . , e 1
2
= 0
n be all block idempotents of FG.
Then, by
Theorem 2.4, a
S(Z(FG))P ei
a
a
Since J(Z(FG)lp = J(Z(FG)IP e l A
+
...
=
...,nl
for all i E {1,2,
0
a + J(Z(FG)IP e n ,
the result follows.
.
weaker version of Corollary 2.5, namely the equality
a+lJ(Z(FG))( p "'(p-') was established by Passman (19801. 3.DEFECT GROUPS OF COVERING BLOCKS.
Throughout this section, G and B
denotes a finite group, N
F an arbitrary field of characteristic p > 0.
i s a block of
group of b .
FG
covering a block
a normal subgroup of
G
Our aim is to prove that if
b of FN, then 6 ( B )
f?
N is a defect
465
DEFECT GROUPS OF COVERING BLOCKS
Let B = BCel
3.1. LEMMA.
b = b(f1
covers b .
tively, such that B (i) V
and
be blocks of FG and FN, respec-
Then the following conditions are equivalent:
is an indecomposable FNrmodule such that V
UN for
B.
U lying in
some projective indecomposable FG-module (ii) V
is a summand of
is a projective indecomposable FN-module such that V
lies in bg = g - l b g
for some g E G. Let X
Proof.
Let f * be the sum of all distinct G-conjugates of f.
(ii), respectively. Then
Y denote the classes of FN-modules satisfying (i) and
and
is a central idempotent of and f*V = V
f*U = U for U E B there is a g
so V E Y
E
G such that
and thus X
To prove that Y
YV
L
5 X,
U E B,
WN.
is a summand of
Then
so L
71
for
V
V,
so
=
E
X. V
is a summand of
(see Lemma 3.10.1).
Because
E
bg.
first assume that
V
E
Hence
V is indecomposahle,
Clearly, V
is projective,
V f E X.
V'
If
is such that L = V/J(FN)V
Y
is a summand of
Let
71
:
uN
for the
No and
then for a suitable r E
r Q - + J(FNl U be a projective cover of J(FN)pU.
But Q E B ,
(&/J(FN1QIN.
Vi/J(FN)Vi for some i and V
Now assume that
V E Y and V'
E
as well.
V
z
E
=@
Vi, all Vie X,
X. Then
.
Then
if
V and V '
V/J(FN)V is a composition
Applying the above argument twice, we derive Q
Since obviously 'V E X
least one
1
QN
so we may assume they have a composition
factor Q/J(FN)Q in common (for some & E Y ) factor of Q
hence
X are in the same block.
are linked (see Proposition 1.10.14),
X2 Y.
e
induces a surjective homomorphism
and therefore L
V E X.
=
5 Y.
is a composition factor of some projective module
FG and ef*
V
E
x
and X
.
is nonempty, X
E
X and
contains at
V in each block b g , so X contains all V in all b g , that is
So the lemma is proved.
We are now ready to prove the following result.
466
CHAPTER 8
3 . 2 . THEOREM. (Knb;rr (19761I.
Let F
be an arbitrary field of characteristic
p > 0, let N be a normal subgroup of
covers b .
and FN, respectively, such that B then D
nN
G and let B and b be blocks of FG If D i s a defect group of B ,
is a defect group of b .
Proof.
6CB)
We denote by
6 ( b ) defect groups of blocks B
and
Let X and Y be the classes of FN-modules satisfying (i1 and
respectively.
We know, from Lemma 3.1, that
(ii), respectively, in Lemma 3.1.
Theorem 3.16.14, there exists an irreducible module M E b where Vz(M) denotes a vertex of M . Therefore M
i s a summand of
UE B
FG-module
and b ,
. Owing
Now M
(U/J(FNIU)N
2
V/J(FN)V
x
=
Y.
By
such that ' U s ( M ) = 6 ( b I , G for some V E X = Y.
for some projective indecomposable
to Lemma 3.16.13, the indecomposable FG-module
U/J(FN)U
is FN-projective, so by Lemma 3.16.1,
On the other hand, by Proposition 3.10.6, there i s a block
6(B]
and
C
1:
SO
6(Bl)
and
6(b) = N n 6(Bll.
B
of
FG such that
Hence
G
6(b) = 6 ( B ) n N , G
as asserted.
4. REGULAR BLOCKS.
Throughout, G
denotes a finite group, H
field of characteristic p > 0, classes of
G and, for each C
is a block of FH, we write potent of FH
lying in b
associated with b .
usual, CRCG)
As E
a subgroup of
CR(G) ,6(C)
G and F an arbitrary
denotes the set of conjugacy
is a p-defect group of
b = b [ e , h ) to indicate that e
and
h
C.
If b
is a block idem-
the irreducible representation of
Z(FH)
We denote by
the natural projection so that T(c+)
Note that
71
=
(c n H I +
for all C E CR(GI
i s a surjective F-homomorphism, but need not preserve multiplication
467
REGULAR BLOCKS
Let
b=b(e,AI
be a block o f
bG i s defined.
LEMMA.
bs
5s 5 G
being a r i n g
corresponding t o
bG o r
i s defined and e i t h e r one of
AG
proving t h a t
b(e,A)
(bSIG i s defined, then so is
C E CR(G1,
2
1
' 1Z ( F S )"1,Z (FHI
we have
.
Hence = (lor
bc
= AoTI =
)on
2
AG
1
( b S ) G i s defined.
i s defined i f and only i f
hG a r e t h e i r r e d u c i b l e r e p r e s e n t a t i o n s of
Z(FG)
Since
(ASIC
a s s o c i a t e d with blocks
b G , r e s p e c t i v e l y , t h e a s s e r t i o n follows.
(ii) By hypothesis,
AG
is an i r r e d u c i b l e r e p r e s e n t a t i o n of
sition 2.2.6, there exists
x(L+) # 0
Proposition 2.2.6,
a s asserted.
=
6 ( b J C_ b ( b G ) . G
(ASf
(bSIG and
b
Consider t h e sequence of n a t u r a l p r o j e c t i o n s
TI = II 71
and t h e r e f o r e
and l e t
.
(bSIG= bG
(i)
Then, f o r a l l
G
be a chain of subgroups of
Z( F G )
Hence
i s an
6 ( b ) always denotes a d e f e c t
I n what follows
bG i s d e f i n e d , then
Proof.
and
FG
1'
r f t h e map
FH.
t h e o t h e r and (ii) I f
H
Let
be a block of If
hen.
b.
group of
(il
=
bG t h e unique block of
homomorphism), we denote by
4.1.
hG
and l e t
Z(FG) (which is e q u i v a l e n t t o AG
i r r e d u c i b l e r e p r e s e n t a t i o n of
and say t h a t
FH
.
f o r some
C E CE(GI
with
6(Cl = 6(b
L E CR(H)
with
L
6 (b) c 6 (L)
i7
G
.
5 c.
)
Then
Z(FG).
such t h a t
6(L)
+ A (C ) # 0 .
56(C) G
Thus
By Propo-
G
and, by
468
CHAPTER 8
For f u t u r e use, we next record 4.2. LEMMA. of
(il
FG
and
B
N
Let
FN,
respectively.
V E B,WE b
W
and
B
Furthermore, i f
be blocks
Then t h e following conditions a r e equivalent:
FG
b,
( i )* (ii) :
W,
and
r e s p e c t i v e l y , such
VN.
i s a submodule of
covers
V
and FN-modules
V
then f o r any i r r e d u c i b l e FG-module
V N
t h e r e e x i s t s an i r r e d u c i b l e submodule of Proof.
b
and
b
covers
(ii) There e x i s t i r r e d u c i b l e
that
G, and l e t B
be a normal subgroup of
B
Write
Ee # 0.
B(E)
=
B,
in
b.
belonging t o
b = b ( e ) so t h a t , by Lemma 3.10.2,
and
If
e
a n n i h i l a t e s a l l i r r e d u c i b l e submodules of
f o r any i r r e d u c i b l e FG-module
V
in
( i )i s equivalent t o
B,
then
e
annihilates a l l irreducible
Ee
B
and so
VNI
B.
FGmodules i n
Hence t h e same i s t r u e f o r
S(B).
potent i n
Since J ( B 1
E
Ee Ee
i s n i l p o t e n t , we conclude t h a t
i s an idem=
0.
Thus ( i )
implies ( i i ) , ( i ):
(ii)
Assume t h a t ( i i l holds.
Then, by C l i f f o r d ' s theorem,
VN = g W C B 1=
g ,g2,...,g,
EeV
=
f o r some
eV = W
Assume t h a t
containing
-1 e = g elg
.. . @ egn W
b
and l e t
g E G.
g-lW i s an i r r e d u c i b l e submodule of
4.3. LEMMA.
H.
If
Let
C ( P ) C_ H
G
then
:iomomorphism with r e s p e c t t o Proof.
eW
and
=
W
and so
( d i r e c t sum of FN-modules)
VN
B
and by
b
covers
w=9-b V N
b = b ( e , X ) be a block of
2 N(P), c
V
b
= 1
b (e 1
)
B.
t h e block of
1
and hence by Lemma 3 . 1 0 . 2 ,
I t follows t h a t
e(g-lW) = g-le so
=
V be any i r r e d u c i b l e FG-module i n
Then, by t h e above
f o r some
EV
and hence t h a t ( i i ) implies ( i ) .
covers
W.
r4
By hypothesis,
W any i r r e d u c i b l e submodule of
Denote by FN
B
kr @
eg
Ee # 0
This proves t h a t
G.
in
... CB gn
P.
XG
= X.p,
#
o
belonging t o
FH where
In particular,
I t s u f f i c e s t o show t h a t f o r each
and l e t
b. P be a p-subgroup of
p : Z(FG)-+
b
G
,
Z(FH) i s t h e Brauer
i s defined.
C E CR(G),
REGULAR BLOCKS
469
XG(C+I = ( l o p ) CC*)
n
C
To t h i s end, w r i t e
fl H =
Ci with each
U
i=1
n(C
+)
=
Ci i n
Ck(H1.
Then
n +
c ci
i=1 and
and so we need only v e r i f y t h a t
xcc3 z Because e i t h e r
c2. -c
L E Ck(H) n suppe
x(cc n c G ( p ) ) +
for a l l
i E {l,.
Ci n CG(P) = 8, we may assume t h e l a t t e r .
C (P) o r G
6(L) = 6(b).
be such t h a t
6(Ci) c 6(L1
Hence
=
Then
P
5 6(L)
h(C)
and s o , by P r o p o s i t i o n 2.2.6,
=
since
0,
Let
P a H.
as r e q u i r e d .
H N
Let
be r e g u l a r w i t h r e s p e c t t o
4.4.
of
Let
LEMMA.
FG.
(i)
G.
be a normal subgroup of
N
N
be a normal subgroup of
is said t o
G
and l e t
B
=
B(E,A)
be a block
B i s r e g u l a r with r e s p e c t t o N
b
of
FN
(iii) B = b'
f o r some block
b
of
FN
Proof.
( i )* ( i i ):
Let
Owing t o Lemma 3.10.4,
such t h a t
G
b = b(e,pl
:
for a l l
FN
which i s covered by
y E FN,
C
h (Z*( F N ) )
z E Z* ( F N )
.
From t h e d e f i n i t i o n of
p
G , we
hence
G
C E CL(G) i s such t h a t
be a block of
p (Z*(FA4 1-
Jt(!.~(r) 1 = h(x1 f o r a l l
p (y) = p ( y )
B
which i s covered by
t h e r e e x i s t s an F-algebra isomorphism
Jt
If
FG
of
Then t h e following c o n d i t i o n s a r e e q u i v a l e n t :
f o r every block
have
B = B(E,h)
.
if
( i i ) B = bG
B.
A block
.. , n l
$ ( p (r))=
A (x)
N,
p (C
then
G +
G J t ( P ( z l ) = A(zl
for a l l =
+
0 = h(C ).
x E Z*(FN)
Hence for a l l
z E Z(FG)
410
CHAPTER 8
and so pG
E i s a block idempotent of bG and thus B
=
b
G
Because p ( E l = h ( E ) = L
Z(FG),
is an irreducible representation of
G
.
(ii) =* (iii): Obvious
G
(iii) * (i): Let b = b ( e , p ) be a block of FN and
such that B = b ,
are equivalent and so it suffices to show that p
C E CR(G1
.
c
with
N.
the result follows.
G +
(c
)
= 0
Then l.I
G
for all
The latter being a consequence of the definition of p
G
,
We have now come to the demonstration for which this section has been developed. Let N
4.5. THEOREM. of FG
and F N ,
(i) If B
be a normal subgroup of
G and let B
and b
be blocks
respectively
is regular with respect to
N, then B covers b if and only if
G
B = b . (iil If D
is a defect group of B
and
CG(D)
5N ,
then B
is regular with
n
respect to N ,
and B
(iii) The block bG
=
B’
is defined if and only if there exists a block of FG
is regular with respect to N
G
E N
for some normal p-subgroup P
Furthermore, by Lemma 3.10.2, if
then bG = bG 1
2
(il
Proof.
if and only if bl If B
assume that B = b potent
E
G
of FG.
covers b ,
and b
bl and b
.
are blocks of F N ,
then B = bG by Lemma 4.4(ii).
G
Conversely,
for some block idem-
for all r E Z*(FN) , we infer, from
Because p (r) = p ( x )
(ii) Write B = B ( E , X )
covers b . and assume that C E C!L(G)
then by Proposition 2.2.6,
D
5 6(C)
is such that C
But then g E N
and so C
ZN,
N.
If
which means that g E C G ( D )
G
some g E C.
is regular
are G-conjugate.
G If b = b ( e , u ) , then B = B ( E , p )
L e ma 3.10. 4, that B
x(C+) # 0,
of G, then B
In particular, bG is defined and is the only block of FG
with respect to N. covering b .
is the unique block of FG which is regular
and covers b .
with respect to N C (P)
which
b.
and covers
(iv) If bG is defined, then bG
(vl If
B of FN which is covered by B .
for any block
a contradiction.
for
471
TAE FONG CORRESPONDENCE
bG is defined, then bG is r e g u l a r (Lemma 4.41 and covers b , by
(iii) I f
B
v i r t u e of ( i ) . Conversely, assume t h a t and covers (iv)
b.
Then, by ( i l ,
(v)
B = b
and
Write
G
B = B(E,X)
56(B)
bG and so bG is defined.
C E CR(G)
and l e t
6(B)
.
5 6(C).
P
and hence
a contradiction.
which i s r e g u l a r
B
FG
be any block of
which i s
bG
.
then by Proposition 2.2.6,
G, P
FG
Then, a s w e have seen i n (iii), bG is r e g u l a r ,
b.
r e g u l a r and covers
b
=
bG i s defined and l e t
Assume t h a t
covers
B
i s a block of
56(C). G
P
Because
N.
C
be such t h a t
If
A(C+) # 0 ,
i s a normal p-subgroup of
By t h e argument of (ii), we deduce t h a t C
5N ,
G
5. THE FONG COFfRESPONDENCX. G
Throughout t h i s s e c t i o n ,
F
and
If
H
i s a block of
G,
i s a subgroup of
FN,
then
G
a normal subgroup of
an a l g e b r a i c a l l y closed f i e l d of c h a r a c t e r i s t i c
6 = B (E )
If
N
denotes a f i n i t e group,
p
3
0.
denotes t h e i n e r t i a group of
G(E)
E.
then Z
T G , H : CFG ( H I
(FG)
i s t h e r e l a t i v e t r a c e map. 5.1.
LEMMA.
Let
module such t h a t of
FH,
then
Proof.
H
FH
1T : FG'
8
{1,g2, . . . , g n}
V@
lies i n
V
FH
FG
and
Owing t o Proposition 1.10.15,
put
X =C-
(C n H I .
C
nH = C Then
V
b
l i e s i n t h e block
.
H
afforded by
in
V
.
(1)
J,
G.
Let
I$
and
F,
respectively.
and
be
it s u f f i c e s t o v e r i f y t h a t
$.rr(C+) = W C + ) To t h i s end, w r i t e
If
( d i r e c t sum of F-spaces)
is a l e f t transversal for
t h e c e n t r a l c h a r a c t e r s of
b
G
be an i r r e d u c i b l e F H -
be t h e n a t u r a l p r o j e c t i o n and w r i t e
... @ gn
n
V" = V @ g where
V
and l e t
VG is an i r r e d u c i b l e FG-module.
bG i s defined and Let
G
be a subgroup of
U C 1
U 2
... U C,
for a l l for suitable
Ci
C E CR(G) E Ck(H),
(2) and
472
CARPTER 8
+2,
and so X
= 0,
Let H be a subgroup of
LEMMA.
5.2.
This proves ( 2 ) and hence the result.
by virtue of (1).
H contains the inertia group of W.
irreducible FN-module such that
W(H) and W ( G )
with N C_HC_G, and let W be an
G
Denote by
the sets of all irreducible FH and FG-modules, respectively,
whose restriction to N
W as a summand.
have n
(i) If
V
E W(H),
fl
V-
(ii) The map
?E
then
W(G1
induces a bijective correspondence between the isomorphism
classes of FH and FG-modules in W(H) Proof.
Let S denote the inertia group of
respectively.
W
and assume that the result
Then, for X E W(S1,
is true for S instead of H.
XI-+f
and W ( G ) ,
the maps
satisfy ti) and (ii), with respect to W ( S ) ,W(H) and
,pectively.
($IG
Because
Hence we may assume that S
WW). Because
V E
v
there exists a composition factor X Since XN
in particular, W of XN
XN.
Then V
W.
of
$ such that
is completely reducible, VN
i s a summand of
isomorphic to
and
W(S1 , W ( G ) ,
res-
Let U
2U
=
H. (#IH,
is an irreducible submodule of "
factor of XH.
9
XG, it follows that ti) and (ii) are satisfied
2
with respect to W(H) , W ( G ) . (i) Suppose that
X*
v
is a composition i s a submodule of
XN
i
denote the sum of all submodules
and, by Clifford's theorem, U
i s an
n
irreducible FH-module such that X that X
2
v".
V is irreducible, we conclude
Since
is also irreducible.
(ii) Assume that V ,V E W(H1 1
( 0, let N of G; let ti1 (iil
If b If
of FH, that b
B
= B(E)
be a block of FN
is a block of
B is a block of FG covering B, called the Fong correspondent of B covers B
and
B =
be an
be a normal subgroup
and let H = G(E).
FH covering 6, then bG
G b .
Let F
is defined and covers
then there exists a unique block in FH
(with respect to
In particular, if b l ,
...,b
8)
B b
such
are all distinct
475
THE FONG CORRESPONDENCE
blocks of FH covering B,
then 6',
... ,bf
are all distinct blocks of FG
8.
covering
(iiiI If b
b(eI
=
is the Fong correspondent of
B
=
B(E1
in FH, then
(el and e is the unique block idempotent of FH with EeE # 0 GtH (b) The restriction of T to 2 ( h ) is an isomorphism Z ( b ) Z ( B ) of FG,H algebras. (a) E
=
T
M,(b),
(c) B
where n
V h VG
(dl The map
=
(G
:
HI
induces a bijective correspondence between the isomorphism
classes of irreducible modules in b (eI The blocks b (f) B
and B
and B ,
respectively.
have a defect group in common
is the unique block of FN covered by
b.
is p-solvable and N = 0 ,(GI, then e
(g) If G
P
H is a defect group of b .
and a Sylow p-subgroup of
= E
In particular, by Step 1 in the proof of Theorem
3.10.9,
b
P
for some k 2 1 not divisible by
Mk(P(H/N))
(i) Write b
Proof.
covers 8 ,
Because b
submodule W
of
=
fl
summand.
(ii) Let B
VN belonging to 6,
W
is contained in H .
bG is defined and
fl
E
9-l) and
Invoking Lemma 5.2,
G
lies in b
has
.
we
W
as a
Therefore,
bG covers 8 . be a block of
such that W
FG covering 6 and let V be an irreducible F G there exists an irreducible FN-module W
Owing to Lemma 4.2, i s a summand of
contained in H , V
L
VN.
V
is defined and
Moreover, by Lemma 4.2, If EeE UG = 0 ,
Since the inertia group of
for some irreducible FH-module U
VG
N has W as a summand (Lemma 5 . 2 ) . then bG
Now, for any g E G, 9W E 6(g
Is an irreducible FG-module, whose restriction to N
module in B .
B
in b .
V
it follows from Lemma 4.2 that there exists an irreducible
Hence, by Lemma 5 . 1 ,
by Lemma 4.2,
and some aEZ*(ff/N,F*).
and fix an irreducible FH-module
b(e1
therefore the inertia group of infer that
p
b
then e E
lies in b
If b
G
covers 8 .
8= 0
and so
,
W is
whose restriction to
is the block of
FH containing U,
by Lemma 5.1, and hence B = b
Let
in
G
.
E be the block idempotent of B .
eEU = E e U
= EU =
0.
But then ~ k = r 0,
CHAPTER 8
476
a contradiction, and hence EeE # 0.
This proves (ti), by appealing to Lemma 5 . 3 .
(iii) Properties (a), (b) and (c) are direct consequence of Lemma 5.3. and b
(a), note that by (c) B ducible modules. where
have the same number of nonisomorphic irre-
f1 P fl2
Thus, it suffices to show that
Vi is an irreducible FH-module in b , i
implies
(VilN, i = 1,2.
since
w = gw
5.2 we have
V
%
1
the same defect.
E =
B1
56(B)
Hence
8@
H.
E
g-ll =
(f)N
and,
B(E)
Consequently,
(V1)N and (V2)N have
contains the inertia group of FJ2,
and, by (d) and Theorem 2.2.3,
6 ( b ) = b ( B ) , proving G
is another block of FN
= B1(E1)
h - l ~ h for some h E H.
conclude that
P
so by Lemma
2
G If
(f)N
such that
V , proving (a).
6(b)
By Lemma 4.1,
But H
then
2
f3
in
we infer that
E
for some g E G, and hence for g E
W 2 as a common summand.
1
($IN,
( V $ l N is a submodule of
8z f,
If
v2,
v1
= 1,2.
Owing to Lemma 4.2, there exists an irreducible FN-module Wi
Wi is a submodule of
To prove
E
1
=
E,
Because
proving (f)
6 ( b ) and
6(B)
have
(el.
covered by b ,
then by Lemma 3.10.2,
is a central idempotent of FH, we
E
.
Property (g) is a consequence of a general result to be proved in the next
.
section (see Proposition 6.4(iii) , (ivl) 5.5. COROLLARY.
p > 0,
Let
F be an algebraically closed field of characteristic
exists a subgroup H of defect group D
G containing 0 ,(GI P
and a block b
Then there
of FH
with
such that where n = ( G : H)
(i) J ( B ) P M n ( J ( b )1 ,
(ii) D
be a block of FG.
let G be a p-solvable group and let B
is a defect group of
(iii) t ( B ) = t ( b ) , where
B
t(B1
and D E Syl ( H I
P
and t ( b ) are the nilpotency indices of J ( B )
and J ( b ), respectively. (ivl b
I-
Mk(f(H/O
P
,(G))
for some
k
1 not divisible by p
and some
a E Z2(X/O ,(GI ,F*). P Proof.
Property (iiil
is a consequence of (i), while properties (i), (ii),
477
TEE FONG CORRESPONDENCE
(c), ( e l , (gl
and ( i v ) follow from Theorem 5 . 4 ( i i i l ,
.
Another u s e f u l observation is provided by 5.6. PROPOSITION. p > 0,
G
let
that
Z(FG)
E E
a block
B
Assume t h a t
1.
.
B
B
covers a block
5
with d e f e c t group
t(B)
t ( & ,ID1
=
B(E)
=
5
with
PO ,(GI P
of
such
0 (5)# 1 and P
with
such t h a t
101
and
5 E Syl
P
and S t e p 1 i n i t s p r o o f , we have
By Theorem 3 . 1 0 . 9 ( i i )
B
=
FG
be a block of
Then t h e r e e x i s t s a p-solvable group
FZ
of
Proof.
be an a l g e b r a i c a l l y closed f i e l d of c h a r a c t e r i s t i c
be a f i n i t e p-solvable group and l e t
D #
d e f e c t group
F
Let
g
Mk
C f l (G/O P ,(GI 1
and
D E Syl (GI P k
f o r some
p
1 not d i v i s i b l e by
(2)
a E Z2(G/0
and some
P
(GI , F * ) .
It is a
standard f a c t ( s e e Karpilovsky (1985, pp.96,97)) t h a t t h e r e i s an e x a c t sequence
where
5-
Z
1-
FE.
f a c t o r of
Hence, by (11, B
Mk(FE) where
B1,
...,Bs
1k ,
p
= M k (E)
5,
-
B.
i s p-solvable, and of
o
P
(G/O
-
such t h a t
1
(3)
f(G/O
Mk(B1l x
... x M k ( B s )
FE,
we i n f e r t h a t
f o r some block
B
and
P
,( G ) )
Mk(F&
is a d i r e c t Since
(4)
FZ
of
%modules coincide.
then by ( 2 ) and (31, ID1 =
d e f e c t group of
x
(GI
i t follows from ( 4 ) and Proposition 1.5.3 t h a t t h e p - p a r t s of t h e
dimensions of i r r e d u c i b l e group of
P'
i s a d i r e c t f a c t o r of
a r e a l l blocks of
B Since
2
i s a c e n t r a l p '-subgroup of
Z
G/O
P
,(G))
101.
in
X 4
1 we have
5
If
X/Z
i s a Sylow p-sub-
0 (G/O , ( G ) ) # 1. P P
t(B) = t(E).
5
is a
Since
G
Hence, t h e i n v e r s e image
satisfies:
and
5
Hence, by Theorem 2.2.7,
Furthermore, by ( 4 ) , w e a l s o have
D #
'
i s a n o n t r i v i a l p-group
CHAPTER 8
470
Z
Since
-
G,
is a c e n t r a l p'-subgroup of
.
follows.
2.
P of
n o n t r i v i a l normal p-subgroup
X
we i n f e r t h a t
Thus 0
P
(2) #
= Z x
P f o r some
1 and t h e r e s u l t
6. THE KULSHAMMER'S STRUCPURE THEOREM. Throughout t h i s s e c t i o n , and
F
G
denotes a f i n i t e group,
N
an a l g e b r a i c a l l y closed f i e l d of c h a r a c t e r i s t i c
FN,
a block of
G,
group of
G(E)
then
a normal subgroup of
p > 0.
denotes t h e i n e r t i a group of
= B(E1
is
H i s a sub-
If
E.
6
If
G
then
T G f H : CFG(HI i s t h e r e l a t i v e t r a c e map.
Given s u b s e t s
NG(X,Yl
ZVG)
---*
X,Y
of
FG,
w e put
n NG(Y)
= NG(X1
All t h e r e s u l t s of t h i s s e c t i o n a r e e x t r a c t e d from an important work of Kiilshammer (1981b).
6.1. LEMMA.
b
and l e t
B
Let =
=
B(E)
be a block of
FG
b ( e ) be t h e Fong correspondent of
( i ) For each d e f e c t group
6
of
6,
B
covering a block
B
= B(E)
d
FN
8.
with r e s p e c t t o
t h e r e i s a d e f e c t group
of
of
b with
6 = d n N (ii) I f
6 =
6
B and d i s a d e f e c t group of
i s a d e f e c t group of
d n N, then f o r each normal subgroup P of
y = ~ ( € 1 of Proof.
F(PN)
(PN:N) = (P:(Pfl N))
lie in
FN
block
y = Y(E)
where
H = G ( d ,
C5N
(bl
C
5P
t h e r e i s a block
.
5 SuppE
i s a power of
by Theorem 2.2.4.
t o o , and w e a r e l e f t t o v e r i f y t h a t
(a)
6
( i ) This i s a p a r t i c u l a r case of Theorem 3.2.
(ii) Since
F(PN)
with d e f e c t group P
d with
b with
of
F(PN).
such t h a t
Hence
E
p,
a l l block idempotents of
is a block idempotent i n
F(PN),
P i s a d e f e c t group of t h e corresponding
Owing t o Proposition 2.2.6,
there exists
C E CR(H),
479
THE K ~ ~ L S H A M M E R STRUCTURE ~S THEOREM
d
(c)
i s contained i n a d e f e c t group of
P
By ( b ) and ( c ) ,
PN = AN.
C.
a
i s contained i n a d e f e c t group
y
of
6
On t h e o t h e r hand, Theorem 3 . 2 i m p l i e s t h a t
=
and so
= Y(E),
A n N.
This g i v e s
the equality
P is
and t h u s 6.2. of
PROPOSITION.
FN
6
b with
6
TI :
B = B(E1
Let
FG
be a block of
8
i s a d e f e c t group of =
= YCE).
D n N.
P
Let
8
covering a block
b = b ( e ) be t h e Fong correspondent of
and l e t
Suppose
y
a d e f e c t group of
B
D
be a normal subgroup of
8.
with r e s p e c t t o
B
D i s a common d e f e c t group of
and
= 8(E)
with
and
6 C P , let
be t h e n a t u r a l p r o j e c t i o n and s e t H = G ( E ) . Then G There i s a block 6 ' = 6 ' ( e ' l of FN(P1 with d e f e c t group D and
FG-+
(i)
FC(P)
H n(e)e'
=
el.
(ii) For each block
D
d e f e c t group
and
6'
i n Ci), t h e r e are blocks
8'
= 8'(€'1,
E ' = TI(€)
,
B' = B ' ( G ' ) FN(PI
of
FN(P) G
of
with
6
with d e f e c t group
N such t h a t (a1
b'
(bl
TI(E)E' = E ' , Proof.
(6')'
6,
=
P.
(Ell
P.
Y'
F(NN(P)). If
g E N
If
Y
= Y(E)
F(PN)
of
= IT(€)
of
with
F ( f l p f l ( P ) ) = F(PNN(P))
i s a c e n t r a l idempotent of
E'
F(NN(P)) with
F(NpN(P)).
E'
Hence
€'El E
'
# 0.
= E
F(CN(P)).
By Lemma 6.1,
TI('€)
= 'TI(€)
On t h e o t h e r hand, f f
g
and, i n p a r t i c u l a r ,
is an element i n
N(P) with G
6'= 6' ( E ' )
Pn
'TI(€)
= TI(€) 'T(E)
Let
is a
E
and we have a block
Again, by Lemma 6.1, t h i s block has d e f e c t group
g E N ( P ) , then G
(PI.
with
= Y'(E')
Clearly
be a block of
c e n t r a l idempotent of of
(B'IG = B
and
It follows, from t h e Brauer correspondence (Theorem 3.9.21),
t h a t t h e r e e x i s t s a block with d e f e c t group
8'
with r e s p e c t t o
( i ) Owing t o Lemma 6.1, t h e r e i s a block
d e f e c t group
81 = 6
B'
fs t h e Fong correspondent of
=
6.
for a l l =
T(E),
CHAPTER 8
480
and so
E
=
Let
:
FG
'II
proving t h a t
'E,
-+
FCG(D) be t h e n a t u r a l p r o j e c t i o n .
Then
n r ( I r ( e ) )= ~ ' ( ef ) 0 a(e)
and
a(e)
e
=
~ ' ( e =] T f l e 1 where each
+
e
2
+
write
... + e
+
Then
n'(e 1
+
... + n ' ( e n )
~ ' ( e i~s ]a c e n t r a l idempotent i n F(CH(D)). This implies t h e b'
existence of a block I t follows t h a t
b'
e'.
=
b'(e')
b'(e')
=
b'
(ii) A s s u m e t h a t =
1
FYYHCPIl.
as a sum of block idempotents i n
T(e)e'
F(NH(P)).
i s a nonzero c e n t r a l idempotent of
=
F(NH(P)) w i t h n ( e ) e '
of
has d e f e c t group
b ' ( e ' ) i s a block of
D, proving
=
and T ' ( e ' ) g o .
e'
(i).
F ( N H ( P ) ) with d e f e c t group
D
and
Then we have
e r E f = e ' n ( e ] a ( a ) = e'.rr(ee) = e ' n ( e ) = e' B'
and we g e t a block
b'
=
and
=
B'(E')
of
F ( N G ( P ) ) with
b'(e'l
i s t h e Pong correspondent of
B' has
d e f e c t group
0.
= B'(E')
C of
By Theorem 5 . 4 ,
with r e s p e c t t o
By Lemma 4 . 3 , t h e blocks
defined and f o r each conjugacy c l a s s
(c n
B'
E r e ' € ' # 0.
(B'IG
and
G o r H , we have
CN,(P))+E~ + J ( Z ( B ~=I T ( C + ) E ' + J ( z ( B ~ ) )
or
(C n y H ( P ) ) + e '+ s ( z ( ~ ' =) )n ( C + ) e ' + J ( z ( ~ ~ I ) , respectively.
Now
.rr(e)e' = e r implies t h a t
( b r ) H = b.
( B r ) G = ( b ' ) G = bG = Bso
n(E)E' # 0
and t h u s
n(E)Ef = E',
proving ( i i ) .
Hence
R'=B'(E')
(b')H are
THE IdkSHAMMER'S STRUCTURE TEEOREM
The next r e s u l t provides an i m p o r t a n t reduction t o 6.3. PROPOSITION.
FN, and l e t b
of
6
Assume t h a t
b
b(e1
=
FG
be a block of
C,(PI.
8
P
Let
D
and
8
covering a block
B
be t h e Fong correspondent of
i s a d e f e c t group of
6 = D n N.
with
B = B(E)
Let
481
=
8.
with r e s p e c t t o
B
a common d e f e c t group of
CP
6
D with
be a normal subgroup of
B(E)
and
and p u t
H = G(E).
b
(i) There e x i s t blocks of
FNH ( P , b l 1
F ( C G ( P ) ) and
A
(a)
=
B
(bl
NG(P,BIl
=
b
bl
b
Fong correspondent of
=
1
b
1
1
P is a normal eel # 0.
1
1
=
b,
a
=
y = Y(E)
y
t h e r e i s a block
= y (E 1
a l(ell
a
E
= 1
A l (El)
B (E 1
of
1
1
= e
1
Since
, E I E e l # 0.
Because
# 0
= E Ee eE = E Ee e = E Ee I 1 1 1 1 1
of
F ( C G ( P I ) with
E 1
Ee E # 0 and a corresponding 1
1
FNG(P,B1).
S = C ( E ) n CG(P).
G i
ee
By Lemma 6 . 1 ,
1
E Ee 1 1
B
=
1
has d e f e c t
a r e a r b i t r a r y blocks of
F(CpN(P)) with
of
1
Eee
=
the
proving ( i l .
FCPN) with d e f e c t group P .
of
Since
i s a block
FNH(P,bl), r e s p e c t i v e l y , a s s t a t e d i n t h e theorem.
1
Set
H
and
1
Ee
=
Then
S
-H C
s i n c e f o r elements g E S
,
we have
gy = g ( ( y l ) P N )= ( S Y l ) P N= ypw = y 1
Let
c
1
)
such t h a t
By Theorem 5.4, w e may assume t h a t
b = b (e
t h e r e e x i s t s a block
A
1
of
1
( s e e Lemma 3.1.11)
6 ( e ) with r e s p e c t t o b = b (e 1
FflH(P,bl).
Assume t h a t
block
D
F ( C H ( P ) ) with
FflH(P,bl) l i e s i n F C H ( P 1
while by Theorem 4.5,
we g e t a block
1
w e may assume t h a t
bl(ell be a block of
=
every block idempotent of
F ( C H ( P ) ) and
1
NH(P,bl)CG(Pl
Let
a = a l ( e l ) of
1
B 1 = B (E )
with d e f e c t group
1
and
G.
subgroup of
(ti1
bl ( e l ) i n ( i l , t h e r e a r e blocks
of FNG(P,B CG (PI (bl) = B1
1
a = a (e
and
D.
(E 1
= A
1
=
H bl = b
F ( C H P l ) with
of
(1) Owing t o PropositTon 6.2,
Proof.
D,
bl(el)
with d e f e c t group
(ii) For each block
group
=
= c l ( f l ) be t h e Fong correspondent of
B
1
=
B ( F - 1 with r e s p e c t t o 1
1
482
Y1
CHAPTER 8
=
Y1(Ell and c
respect to Y
=
c,lf,)
= y1 (El).
the Fong correspondent o f
1
and for elements g E C,(P)
- S,
with
f, = f,.
We wish to show that
Assume by way of contradiction that f
b 1 = b1 (e 1 )
+ f2.
For elements g E S ,
we have
we have
Thus
contrary to
# 0.
Ee E
E
1
1
Hence f
=
f, and
1
CGP)
C,(P)
=
bl
c1
B1
=
It follows that
and A
1
= A (E 1
1
has defect group D.
NOW (a) follows from
bG = bG = B 1
B1
It follows from (a1 that N H ( P , b )CG(P1 5 NG(P,B1). tainment, assume that g
€
NG(P,B1l.
To prove the reverse con-
Then
O # g ( ~ E )= g ~ E 1 1
By Lemma 3.10.2,
1 1
there exists an element c E C G ( P ) with
= E
"E
1
particular, gc E H . of
Hence "5' = S and " c
= gccl (gcfl)
FS with 0 # gc(ElflEl)= E gcf 1
E
1 1
The uniqueness of the Fong correspondent implies that
gCf which in turn implies that
1
=
f
1
and "c
1
=
c
1
1
,
in
is a block
48 3
Let G
6.4. PROPOSITION.
a block of FG
FN with
be p-solvable, let hr = 0 , ( G I
with defect group D
natural projection.
such that B
Put P = D n O
EEZ(FG).
(GI
P'rP
6
covers a block
be
= 6(E)
andlet n : F G + F ( C G ( P ) l
of
bethe
Then
P is a Sylow p-subgroup of 0
(il
and let B = B ( E )
P
(G)
P'rP
such that
0 ,(NG(P1) = "(PI
P
and 0 ,(NG(D1l = NN(D1
P
(ii)
(iii) E
F(CN(P)l ,F(CG(Pll and F(NG(P)l.
is a block idempotent of
T(El
= E
D is a Sylow p-subgroup of G
(iv)
Proof.
(il
Theorem 2.2.4,
Choose a Sylow p-subgroup E
y
=
g-lDg.
P'IP
(G)
with P
of F(QN) has defect group Q.
= Y(E)
there is a defect group d
of 0
of
B
with Q
5 Q.
By
F(&N), and by Theorem 3.10.9,
is a block idempotent o f
corresponding block
that d
Q
=
d
0
P'rP
(GI.
the
By Lemma 6.1,
Let g E G
be such
Then
Pc Q
=
g-log n
o
(G) =
PrrP
g-lpg
and
Hence P
is a Sylow p-subgroup o f
0 ,(N P)l
C
P
6
ICG(Pl 5 PN.
0
P'rP
(GI.
It follows from Lemma 7.1.3
Taking into account that N G ( D )
5 NG(P),
that
(i) is
established. (iil As we have seen in (i), the block By Proposition 6.2, we get a block
y'
of F(PN) has defect group P.
y = y(e)
= y'(E'),
E'
= T(E),
Of
I" (P)) = F(PNN(P))= F(PCN(Pl), a block 6 ' = B ' ( E ' ) of F ( C N ( P ) ) and a block B' = B ' ( E ' ) of F(N(P)) with E ' E ' # 0. By Lemmas 5.3 and 3.1.11,
F(IV
G
E f E F(CG(p)) 5 F(PN) and even E' E F(PCN(P)). So we get E'
=
E'E' =
E'
and
484
CHAPTER 8
( i i ) follows
B1
(iii) L e t
=
B1
d e f e c t group of of
0
P'rP
B 1 (El) be a block =
FG
of
with
As i n ( i ),
B1 ( E l ) .
(GI and so w e may assume t h a t P 0 # T(E
so (ii)implies t h a t
= 1
n ( E 1 = IT(€).
n(E
El
# 0, and l e t
E
D1 n 0
P'tP
(G)
D n 0
=
P'PP
1
= IT@
E)
D
be a
1
i s a Sylow p-subgroup
(G).
I t is clear that
)IT(€),
Thus
1
0 # IT(E )IT(E) = n(EIE) 1
E
and (iv)
E,
= 1
.
proving ( i i i ) .
D i r e c t consequence of (iii)and Theorem 3.10.9(11).
The next observation is t h e so c a l l e d F r a t t i n i argument.
6.5.
N.
group of
Then
Proof.
g-lPg
N
Let
LEMMA.
G = I\IG(P1*N
g be an a r b i t r a r y element of
Let
i s a Sylow p-subgroup of
.
n E N.
some
required. 1
Let
+
a
4 X+
%
Y
-t
Then
and hence
5N
g-'Ng
and so
g-lPg = n - h
Hence, by Sylow theorem, G
for
g E NG(P)*N, a s
Z
be an extension of f i n i t e groups, where
1
8 i n t h e following way:
:
For any
Y-
Out
is not
z = AutZ/InnZ
y E Y,8(y)
X
ranges over those elements i n
3:
G.
This extension d e f i n e s a homomorphism
n e c e s s a r i l y abelian.
where
N.
gn-l E N (PI
I t follows t h a t
2
G and l e t P be a Sylow p-sub-
be a normal subgroup of
c o n s i s t s of a l l maps f o r which
=
+ Z , Z H , ' ; Z
y.
W e come now t o t h e main r e s u l t of t h i s s e c t i o n . 6.6.
(Kclshammer (1981b).
THEOREM.
G be a f i n i t e p-solvable group, l e t F
Let
be an a l g e b r a i c a l l y closed f i e l d of c h a r a c t e r i s t i c
FG
block of blocks
B
= B (E 1
group (i)
G B
D =
with d e f e c t group 1
)
of
F ( C G ( P ) ) and
= A
A 1
H of
N (P,B )
G
and l e t
B = B(E)
Then t h e r e a r e a normal subgroup
1
and a subgroup
B
D.
p,
I
1
(E 1
of
FNG(P,B )
be a
P of
D,
with d e f e c t
1
such t h a t t h e following p r o p e r t i e s hold:
THE K~~LSHAMMER'S SPRUCTURE
485
THEOREM
is a Sylowp-subgroup of H
(ii) D
(iii) Po , ( H I
P
(iv) B
1
(v) H/O
= 0
P'rP
Mn(FCY(H/6
P
(H)
,( H I )
1 and some a E Z2(H/0 , ( H ) ,F*)
for some n
P
is isomorphic to an extension of P with the group NG(P,B1)/PC(P),
,(H)
P
G
where the corresponding homomorphism
is induced by the natural homomorphism NG(P,BI)-+ Au-. Put N = 0 ,(GI
Proof.
P
and let 8
We argue by induction on the order of
Case 1. Let b
S =
=
FN
be a block of
= B(E)
covered by
B.
G and distinguish two cases.
G(E) # G.
b ( e ) be the Fong correspondent of B
with respect to
5.4, we may assume that D is a common defect group of B there are a normal subgroup P o f
D, blocks b
=
6.
By Theorem
and b .
By induction,
b l ( e l ) of F ( C ( P ) ) and S
a
= a l ( e l ) of
FN (P,b 1 with defect group D, a subgroup H of N S ( P , b l ) ,
s
a natural number rn
1
and
CY E
Z2(E,F*I where
=
H/o P , ( H I ,
such that the
following conditions hold:
b
S
= b
D is a Sylow p-subgroup of H
is isomorphic to an extension of P where the automorphisms are induced by N s ( P , Put n = rn(G of FN(P,B
G
and choose blocks B
: S)
as in Proposition 6.3.
1
)
/C,(P)
= B (E ) 1
1
of
. F(C(P)) and A G
=
A1(EI)
Then (i) is a consequence of (i') and
1
Proposition 6.3, (iv) follows from Theorem 5.4, and by Proposition 6.3,
NG(P,B1)/C(P) N(P,bll/C(P) G S S which implies (v).
Case 2.
S = G(E) =
In this case, E
G
is a central idempotent of FG, so we may apply ~roposition6.4.
486
CHAPTER 8
P
Put
op ,, p (GI , H
D n
=
= N~ ( P I
,
F(CG(P)
FG
:
'TI
be t h e n a t u r a l p r o j e c t i o n .
and l e t
Denote by
B1
=
B (E 1
1
)
A
and
F(CG(P)) and F(NG(P)) with t h e block idempotent E
of
N (P1 = H = NG(P,B ) .
By Proposition 6 . 4 ( i i ) , w e have block
A
=
Al (El)
(iii) a r e now d i r e c t consequence of Proposition 6.4.
= A (E 1
t h e blocks
1
respectively.
= 7T(E),
By Proposition 6 . 2 , t h e
G
F(NG(P)) h a s d e f e c t group D.
of
1
P a r t s ( i ) ,( i i ) ,and By Lemma 6.5,
and
G/N
2
NG(P1/NN(P1
= N(P)/CN(P)
G
we have
From Lemma 7.1.3,
CG(P)
5 PN.
Thus (iv) and (v) a r e immediate conse-
quences of t h e isomorphism e s t a b l i s h e d i n S t e p 1 of Theorem 3.10.9.
This
completes t h e proof of t h e theorem.
G
Let
and
G
w e say t h a t
G
H.
on
I f a homomorphism ($ : G - + A u t ( H ) i s given,
be two groups.
acts on H
The a c t i o n
0
Let
H
($
@;
via of
is called f a i t k f u 2
G
be an a c t i o n of a group
product H H G
( o r simply
t h e homomorphism ($
H gG
G
if
on another group
i f the action
($
i s c a l l e d an action of Ker@ = 1.
H.
The semidirect
i s understood) c o n s i s t s of
($
( h , g ), h
a l l pairs
E
H, g E G ,
Usually, t h e subgroups
(H,lI
with t h e product defined by t h e formula
and
(1,G)
a r e i d e n t i f i e d with
H
and
G,
r e s p e c t i v e l y , so t h a t
H X G = HG, H 4 H X G for a l l 6.7. that
ghg-'
=
@ ( g )( h )
g E G, h E H .
COROLLARY.
P
and
=
D.
Further t o t h e assumptions and n o t a t i o n of Theorem 6.6, assume Then
THE K~SHAMMBR'S
n a 1
f o r some
H
Let
H/o
verify that
c1
P
,( H )
D xs D
i s a p'-group
H/Op
s.
and hence
,( H I .
P
S
D
Since
H
=
i s normal.
0 ,(H/o , ( H I ) = 1,
Because
S
f o r a s u i t a b l e p'-group
H/op, (HI
t h e Sylow p-subgroup of
a c t i n g f a i t h f u l l y on
and hence
P,
D
1
acts f a i t h f u l l y on
THEOREM. (Kkshammer (1981bl 1.
Let
D 4 S f o r some
H / o ,(HI P
Thus
D,
a s required.
FG
D.
with d e f e c t group
that
9
P
=
D.
G be a f i n i t e p-solvable group, l e t
be an a l g e b r a i c a l l y closed f i e l d of c h a r a c t e r i s t i c block of
DO , ( H ) / O , ( H ) P P
i t follows from Lemma 7 . 1 . 3
P
D.
Theorem 6.6(v) i m p l i e s t h a t
W e c l o s e t h i s s e c t i o n by e x h i b i t i n g circumstances under which
6.8.
acting
By Theorem 6 . 6 ( i v ) , it s u f f i c e s t o
i s a Sylow p-subgroup of
is a Sylow p-subgroup of
cDs(D)5 D
S
where
be as i n Theorem 6.6.
By Theorem 6 . 6 ( i i ) ,
p'-group
E Z 2 ( D X S,F*),
D.
f a i t h f u l l y on Proof.
and some
487
STRUCTURE THEOREM
p
and l e t
B = B(E)
F
be a
A s s u m e t h a t a t l e a s t one of t h e following
c o n d i t i o n s hold: (a)
G has p-length
(bl
D
(cl
D is a T . I .
1
i s abelian
G
subgroup of
(i.e.
f o r each
g
E G,
either
g
-1
Dg
=
D or
S - l D g n D = 11 For a l l normal subgroups
(d)
BG
Then
P = D,
=
D
of
and a l l blocks
where
P i s a s i n Theorem 6.6.
f o r some
n 2 1 and some a E Z2(D N S , F * ) ,
f u l l y on
D.
Proof. (a)
If
G
BQ
= BQ(EQl
of
F(CG(&)
B , fl (&,B 1 = DCG(&) G Q
with
Q
Q
Hence, by Corollary 6.7,
where
S
is a p'-group
acting faith-
W e keep t h e n o t a t i o n of Theorem 6.6.
has p-length
1,
H
then
i s a l s o of p-length
1.
Hence, by Theorem
6.6 ( i i ) ,( i i i ) ,
DO , ( H ) P and t h e r e f o r e (b)
If
D
=
0
P'IP
( H ) = PO , ( H )
P
P = D.
i s a b e l i a n , then
D
5 CG(P).
I t follows from Lemma 7.1.3 and Theorem
488
CHAPTER 8
6.6. (iii) t h a t
D
CH(P)
(c)
Assume that
D is
and
D
If
= 1 = P.
5 POP ,(HI T.I.
a
P
= D.
subgroup of
G.
P # 1, then
DOp,( H I and so (d)
D
=
H
c_
and
5NG(P1
P
If
~NG(D1.
(H1 0 P'rP
=
Po
P
= 1,
then
H
= 0
Pf
(H)
Hence
, (HI
P.
In t h i s case we have
H = H n D C G ( P ) = DCH(P) C_ DO
(H)
P
by Lemma 7.1.3.
DO , ( H I = PO ,(HI
Thus
and
P
P
D
P.
=
Note t h a t i n t h i s case
S = 1.
even
?. APPLICATIONS.
G
Throughout t h i s s e c t i o n , characteristic
J(B) of
p > 0 and t ( B ) B
of a block
J(FG).
of
FG.
Given s u b s e t s
F
denotes a f i n i t e group,
t h e nilpotency index of t h e Jacobson r a d i c a l
As u s u a l , we w r i t e
X,Y
an a r b i t r a r y f i e l d of
of
FG,
t(G1
f o r t h e nilpotency index
we put
N ~ ( X , Y I= N (XI n N (YI G G 7.1.
Let
LEMMA.
B
a l g e b r a i c closure of group
D
be a block of
F.
FG
with d e f e c t group
Then t h e r e e x i s t blocks
B1,
D
...,Bk
and l e t of
EG
E be t h e with d e f e c t
such t h a t for a l l
m 2 1
In p a r t i c u l a r ,
Proof.
By Lemma 3.16.10,
we can w r i t e
E 8 B = B
F
f o r an i n t e g e r
k 2 1, where each B
Invoking Corollary 3.1.18,
i
@...
@Bk
i s a block of
we t h e r e f o r e derive
FG
with d e f e c t group
D.
APPLICATIONS
as required.
8
Let G
7 . 2 . THEOREM.
be a finite p-solvable group and let B
with defect group D. (a) G
has p-length
(b) D
is abelian
(c) D
is a T . I .
BG = B , IVG(Q,B 1
1
subgroup of G
=
Q
be a block of FG
Assume that at least one of the following conditions hold:
(d) For all normal subgroups Q
Q
489
of
D and all blocks B
Q
of F ( C G ( Q ) ) with
DC,(Q)
Then
Owing to Lemma 7.1, we may harmlessly assume that F
Proof.
ically closed. having D
Hence, by Corollary 6.7,
t ( B ) = t(Pff1 where
as a normal Sylow p-subgroup and CY E 2’ ( H , F * ] .
is algebraH
is a group
Thus t ( B l
=
t (D) by
virtue of Proposition 6.3.8. Let G be a finite p-solvable group and let B
7.3. COROLLARY. FG
with abelian defect group D
... x Z!
Z! x ? lx pnl pn2
, ni
be a block of
> 0, 1 < i
k.
Pnk
Then
t(B1 = 1 - k
k
+
ni
C p i=1
Direct consequence o f Theorem 7.2 and Corollary 3.2.5.
Proof.
8
To prove our next result, we require the following preliminary observation. 7.4. LEMMA.
Let B = B ( e ] be a block of F G ,
G
:
and let f
(i) f ( B )
=
FG-+
@
B 1
(ii) t ( B )
>
t(P)
Proof.
5
E { B 1 ,... ,B
F(G/P)
... d Bn +
t(Bi)
let P be a normal p-subgroup of
be the natural homomorphism.
for some n 2 1 and some blocks B 1
-
1 for all
iE
11,
...,n}.
Part (i) is a consequence of Lemma 2.3. n
)
and put
Then
t
=
t(5).
Let
7
=
of
F(G/P)
To prove (ii), fix
be the restriction of f
Then Ker? = Kerf n B
,...,Bn
(Kerf)@ = I ( P ) B
to B .
CHAPTER 8
490
F(J(B))= J(T(B)).
and, by Lemma 6 . 1 . 1 ( i i i ) ,
J(!ht-l
Since
# 0, w e d e r i v e
Then Ker? = I ( P ) B
J(B)t-l and so t h e r e e x i s t s
G.
s i t i o n of t)
9 I(P)FG
J(Blt-l
t) E
We can w r i t e
= FG*I(P).
-
tJ =
I(P)B.
Let
C 9.s. j=1 3 3
with
4
Thus we may assume t h a t
4
G = U g . P be a c o s e t decompoj=1 3 s FP. Clearly, j s1
9 I(P),
i n which case w e may
write s
'
=
Ccxx,cxEF SLEP
+
W p # 0.
I t w i l l next be shown t h a t
,
Cc2#O
and
6 P
Assume t h e contrary.
Since
+
WP = (Cg .s .)P+ = cg'. ( s .P+, 3 3 3 3 and s i n c e
S
.€"E
Fp
3
for a l l
j,
+
0 = SIP
c c = 0, 31EP
and so
w e have
s .P + 3
=
0 for a l l j .
Hence
c CXX)P+ = ( c c ) P + 6 P XEP
= (
a contradiction.
Thus
I@' # 0.
Since
by Lemma 3.11.2,
and s i n c e
e I ( P ) ~= I ( P ) he h a 0,
f o r any i n t e g e r
5 S ( B )h
we d e r i v e
wp+ E J ( B ) t + t ( P ) - 2 Thus
t(B)
+
t
t(P1
-
1,
as a s s e r t e d .
We a r e now ready t o prove 7.5.
THEOREM.
and l e t (i)
B
(Koshitani (19831, (1984)).
be a block of
d(p-1) + 1 < t ( B )
(ii) t ( B ) = pd
FG
Let
with d e f e c t group
pd
i f and only i f
D is c y c l i c .
G
be a f i n i t e p-solvable group
D of o r d e r p d .
APPLICATIONS
By Lemma 7.1, we may assume that F
Proof.
d = 0
then J ( B ) = 0
is algebraically closed.
If
by Theorem 3.6.4(1), in which case both assertions are
We may therefore assume that d > 0.
trivial.
pa
We first show that t ( B )
t(B1
with
By Corollary 5.5, we may assume that D
t(B)
Proposition 7.2.1,
l ,
[GI.
+
We argue by double induction
1.
If G = 6 ( B ) ,
wehave p I I G I .
then B = F G
and the required assertion is true by Proposition 7.2.1. Let N = 0
P'
(G)
and let 6
=
B(E)
be a black of FN
Theorem 5.4,
FH, H = G(E),
has a block
t(B) = t(b).
If G # G ( E ) ,
then since
induction.
Hence we may assume that G
=
b with the same defect d IG(E)
I
0, let U
be the F-dimension of the projective cover of the be the principal block of FG.
trivial FG-module and let B
If P E Syl CG)
P
is
cyclic, then
.
dimJGB1 = dimB(1- l / u l
F if and only if G
F
is p-solvable of p-length
8.23. THEOREM. miller (1971)).
Let
sn
1.
be the symmetric group of degree n.
Then lim(dimJ(FSnl/dimFSn) = 1
n
F
+ m F
8.24. THEOREM. (Koshitani (1982b)).
Assume that p
=
2 and
G is a 2-solvable
group with a Sylow 2-subgroup P which satisfies one of the following conditions: (l.1
Ciil
P is cyclic of order 4 P
is elementary abelian of order
(iii) G = S Then t(G)
=
4.
.
8
501
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I1984al On the radical of an infinite group algebra, Math. J . Okayama Univ. 26, 13-18.
I1984bl On a theorem of S. Koshitani, Math. J. Okayama Univ. 26, 19-22. DKUYAMA, T.
119781 A note on the Brauer correspondence, Proc. Japan Acad., 54, Ser.A, 27-28. [19801 Some studies on group algebras, Hokkaido Math. J. 9. 217-221. 119811
On the radical of the center of a group algebra, Hokkaido Math. J. 10, 406-408.
BIBLICGRAPAY
515
OKUYAMA, T.
119831
On finite group algebras with radical cube zero, Proc. 16th Sympos. on Ring Theory, Okayama Math. Lect., Okayama Univ. Okayama, 105-111.
119861 On blocks of finite groups with radical cube zero, preprint. OKUYAMA, T and TSUSHIMA Y.
119831 Local properties of p-block algebras of finite groups, Osaka J. Math. 20, 33-41 [1984] Extf; for irreducible modules, Comm, Algebra 12 (18), 2273-2277. 119861 On a conjecture of P . Landrock, preprint. OSIMA, M.
[19421 On primary decomposable group rings, Proc. Phys.-Math. SOC. Japan (3) 24, 1-9. I19551 Note on blocks of group characters, Math. J. Okayama Univ. 4, 175-188. 119661 On block idempotents of modular group rings, Nagoya Math. J. 27, 429-433. PASSMAN, D.S. 119621 Nil ideals in group rtngs, Michigan Math. J. 9, 375-384 11969aI Central idempotents in group rings, Proc. Amer. Math. SOC. 22, 555-556. 11969bl Blocks and normal subgroups, J. Algebra 12, 569-575. [1969c] On the semisimplicity of modular group algebras, Proc. Amer. Math. SOC. 20, 515-519. [1969d] On the semisimplicity of modular group algebras 11, Canad. J. Math., 21, 1137-1145. I1970al Radicals of twisted group rings, Proc. London Math. SOC. (31 , 20, 409-437. [197Ob] On the semisimplicity of twisted group algebras, Proc. Amer. Math. SOC. 25, 161-166. I1973aI On the semisimplicity of group rings of linear groups, Pac. J. Math. 45, 221-228. [1973b] On the semisimplicity of group rings of linear groups 11, Pac. J. Math., 48, 215-234. [1974a] The semisimplicity problem for group rings, Symp. Math. 8 , 333-342. [1974b] Advances in group rings, Israel J. Math. 19, 67-107. [19751 On the semisimplicity of g r x p rings of some locally finite groups, Pac.~. Math. 58, 179-207. I19771 The algebraic structure of *grouprings, Wiley, New York.
516
BIBLIOGRAPHY
PASSMAN , D. S . 119801
The r a d i c a l of t h e c e n t e r of a group a l g e b r a , Proc. Amer. Math. Soc. 78, No.3, 323-326.
PWITER, G. 119771
On t h e r a d i c a l of a group a l g e b r a over a commutative r i n g , Glasgow Math.J. 18, Nol, 101-104.
PUIG, L. 119861
The Source algebra of a n i l p o t e n t block, p r e p r i n t .
PUTPASWAMAIAH, B.H. 119771
and DIXON J . D .
Modular r e p r e s e n t a t i o n s of f i n i t e groups, Academic P r e s s , N e w York, London.
RENAULT, G.
[1971]
Sur les anneaux de groupes, C.R.
Acad. S c i . P a r i s , 273, 84-87.
REYNOLDS, W.F.
[19631
Blocks and normal subgroups of f i n i t e groups, Nagoya Math. J. 2 2 , 15-32.
I19661
Block idempotents of twisted group a l g e b r a s , Proc. Amer. Math. SOC. 1 7 , 280-282.
119721
Sections and i d e a l s of c e n t e r s o f group a l g e b r a s , J. Algebra, 2 0 , 176-181.
RICKART, C.
119501
The uniqueness of norm problem i n Banach a l g e b r a s , Ann. Math. 51, 615-628.
ROBINSON, G.R. 119831
The number o f blocks with a given d e f e c t group, J. Algebra, 84, 493-502.
ROSEN, M . I . [19661
The Jacobson r a d i c a l of a group algebra, Michigan Math. J. 13, 477-480.
ROZENBERG
119611
, A.
Blocks and c e n t e r s of group a l g e b r a s , Math. Z , 76, 209-216.
SAKSONOV, A . I . 119711
On t h e decomposition of a permutation group over a c h a r a c t e r i s t i c f i e l d , Soviet Math. Dokl. 1 2 , No.3, 786-790.
SCHNEIDER, H. I19671
and WEISSGLASS, J
Group r i n g s , semigroup r i n g s and t h e i r r a d i c a l s , J. Algebra 5 , 1-15.
SCHWARTZ, W. 119791
D i e s t r u k t u r modularer Gruppenringe Endlicher Gruppen d e r p-Lange 1, J. Algebra 60, 51-75.
BIBLIOGRAPHY
[19771
517
Linear r e p r e s e n t a t i o n s of f i n i t e groups, Springer, New York, Heidelberg, Berlin.
SINHA, I. I19741
Semisimplicity of group r i n g s with t r i v i a l u n i t s , Tamkang J . Math. 5 , 1, 107-108.
SPIEGEL, H. I19701
D a s Radikal d e r modularen Gruppenalgebra e i n e r endlichen Gruppe, Diplomarbeit, T;ibingen.
119741
Blockkorrespondenzen und p ' - N o m a l t e i l e r , 25, 483-487.
SRINIVASAN,
119601
Arch. Math.
(Basel)
B.
On t h e indecomposable r e p r e s e n t a t i o n s of a c e r t a i n c l a s s of groups, Proc. London Math. SOC. ( 3 ) 10, 497-513.
STONEHEWER, S.E. [19691
Group a l g e b r a s of some t o r s i o n - f r e e groups, J. Algebra 13, 143-147.
SWAN, R.G.
119631
The Grothendieck group o f a f i n i t e group, Topology 2 , 85-110.
TSUSHIMA, Y. [1967]
Radicals of group a l g e b r a s , Osaka Math. J. 4 , 179-182
119681
A group a l g e b r a of a p-solvable group, Osaka Math. J. 5 , 89-98.
[1971a] On t h e a n n i h i l a t o r i d e a l s of t h e r a d i c a l of a group a l g e b r a , Osaka Math. J. 8, 91-97. [1971b] On t h e block of d e f e c t zero, Nagoya Math. J. 44, 57-59. I19741
On t h e e x i s t e n c e of c h a r a c t e r s of d e f e c t z e r o , Osaka Math. J.11, 417-423.
[1977)
On t h e weakly r e g u l a r p-blocks with r e s p e c t t o 14, 465-470.
I1978al On t h e p'-section 83-86.
0 '(G),
P
Osaka Math. J.
sum i n a f i n i t e group r i n g , Math. J. Okayama Univ. 2 0 ,
[1978b] Some n o t e s on t h e r a d i c a l of a f i n i t e group r i n g , Osaka Math. J . 15, 647-653. [1978c] On t h e second reduction theorem of P. Fong, Kumamoto J. S c i . 6-14.
(Math) 13,
[1978d] Problems on t h e r a d i c a l of a f i n i t e group r i n g s , Proc. 10th. Sympos. on Ring theory (Shinhu Univ. Matsumoto, 1 9 7 7 ) , Dept. of Math. Okayama Univ, Okayama, 116-120.
BIBLIOOWHY
518
TSUSHIMA, Y. I19791
Some notes on t h e r a d i c a l of a f i n i t e group r i n g 11, Osaka J. Math. 16, 35-38.
[1983]
A note on Cartan i n t e g e r s f o r p-solvable groups, Osaka J. Math. 2 0 ,
675-679. 119861
On p-radical groups, p r e p r i n t .
VILLAMAYOR, O.E.
119581
On t h e semisimplicity of group a l g e b r a s , Proc. Amer. Math. SOC. 9 , 62 1-627.
119591
On t h e semisimplicity of group a l g e b r a s , 11, Proc. Amer. Math. SOC. 9 , 621-627.
WADA, T.
I19771
On t h e e x i s t e n c e of p-blocks with given d e f e c t groups, Hokkaido Math. J . 6 , 243-248.
119811
Blocks with a normal d e f e c t group, Hokkaido Math. J. 10, 319-322.
[19831
On t h e number of i r r e d u c i b l e c h a r a c t e r s i n a f i n i t e group, Hokkaido Math. J. 1 2 , 74-82.
119851
On t h e number of i r r e d u c i b l e c h a r a c t e r s i n a f i n i t e group, 11, Hokkaido Math. J. 14, N02, 149-154.
WALLACE, D.A.R. P h i l . SOC. 54, 128-130.
119581
N o t e on t h e r a d i c a l of a group algebra, Proc. Camb.
[19611
On t h e r a d i c a l of a group a l g e b r a , Proc. Amer. Math. SOC. 1 2 , 133-137.
I1962al Group a l g e b r a s with r a d i c a l s of square zero, Proc. Glasgow Math. Assoc. 5 , 158-159. 11962b1 Group a l g e b r a s w i t h c e n t r a l r a d i c a l s , Proc. Glasgow Math. Assoc. 5 , 103-108. I19651
On t h e commutativity of t h e r a d i c a l of a group algebra, Proc. Glasgow Math ASSOC. 7 , 1-8.
[19671
The Jacobson r a d i c a l s of t h e group a l g e b r a s of a group and of c e r t a i n normal subgroups, Math. Z. 100, 282-294.
[19681
Lower bounds f o r t h e r a d i c a l of t h e group algebra of a f i n i t e p-solvable group, Proc. Edinb Math. Soc(2), 16, 127-134.
[1969]
On commutative and c e n t r a l conditions on t h e Jacobson r a d i c a l of t h e group algebra of a group, Proc. London Math. SOC. 19, 385-402.
119701
The r a d i c a l of t h e group algebra of a subgroup, of a polycyclic group and of a r e s t r i c t e d SN-group, Proc. Edinb. Math. SOC. V o l , 17 ( S e r i e s I1 P a r t 2 , (165-171).
119711
On commutative and c e n t r a l c o n d i t i o n s on t h e Jacobson r a d i c a l of t h e group algebra of a group 11, J. London Math. SOC. 4, 91-99.
519
BIBLIOGRAPHY
WALLIS, W.D. I19681
A reduction of t h e problem of semisimplicity, J . Algebra 10, 501-502.
WARD, H.N.
119611
Some r e s u l t s on t h e group a l g e b r a of a group over a prime f i e l d , Seminar on F i n i t e Groups an'd Related Topics, Harvard Univ., 13-19.
119681
The a n a l y s i s of r e p r e s e n t a t i o n s induced from a normal subgroup, Michig. Math. J. 15, 417-420.
WATANABE , A.
I19791
On Fong's r e d u c t i o n s , Kumamoto J. S c i .
m a t h ) , 13, 40-54.
I1982al R e l a t i o n s between blocks of a f i n i t e group and i t s subgroup, J . Algebra 78, 282-291. 11982bI p-blocks and p-regular c l a s s e s i n a f i n i t e group, Kumamoto J . S c i . , (Math), 1 5 , 33-38. WILLEMS, W. 119761
Bemerkungen z u r modularen D a r s t e l l u n g s t h e o r i e 111. I n d u z i e r t e und eingeschrgnkte Moduln, Arch. Math. (Basel) , 26, 497-503.
119781
%er d i e Existenz von Blscken, J. Algebra 53, 402-409.
119801
On t h e p r o j e c t i v e s of a group a l g e b r a , Math. Z.
1 7 1 , 163-174.
ZALESSKII, A.E. I19651
On t h e semisimplicity of a crossed product, S i b i r s k . Math. Zh. 6 , 1009- 1013.
[19701
On group r i n g s of s o l v a b l e groups, Izv. Akad. Nauk BSSR, s e r , Fiz-Mat., 13-21.
I19731
On t h e semisimplicity of a modular group algebra of a s o l v a b l e group, S o v i e t Math. 14, 101-105.
119741
The Jacobson r a d i c a l of t h e group algebra of a s o l v a b l e group i s l o c a l l y n i l p o t e n t , Izv. Akad. Nauk. SSSR, Ser. Mat. 38, 983-994.
ZASSENHAUS, H. I19491
The theory of groups, 2nd ed. Chelsea, New York.
52 0
Notation
Number Systems
the n a t u r a l numbers the rational integers t h e r a t i o n a l numbers t h e r e a l numbers t h e complex numbers t h e i n t e g e r s mod
ni
S e t Theory proper i n c l u s i o n L -
inclusion
I XI
t h e c a r d i n a l i t y of t h e s e t
x-Y
t h e complement of
Y
in
x x
Number Theory
a divides b
a does not d i v i d e b g r e a t e s t connnon d i v i s o r of t h e p - p a r t of
n
and
b
NOTATION
n
t h e p'-part
P'
521
n
of
Group Theory
F*
t h e m u l t i p l i c a t i v e group of a f i e l d
<X> 'n G x G 1
2
t h e subgroup generated by
X
t h e c y c l i c group of o r d e r
n
F
GI and G2
d i r e c t product of
H dG
H
D NS
t h e semidirect product of a normal subgroup subgroup S
X
t h e normalizer of
=
[G,Gl
D and a
X in G
t h e c e n t r a l i z e r of
G'
G
is a normal subgroup of
G
in
t h e commutator subgroup of
G
n Gi
d i r e c t product of
? Gi
r e s t r i c t e d d i r e c t product (or d i r e c t sum) of
Gi, i
E
I
iEI
ZEI
G
expG
t h e exponent of
Z (GI
t h e c e n t r e of
G
t h e p-part of
g
the p'-part
g
of
t h e set of a l l conjugacy classes of
G
t h e s e t of a l l Sylow p-subgroups of
G
t h e s t a b i l i z e r of t h e o r b i t of
x
t h e index of
H
z
in
G
t h e maximal normal p'-subgroup of t h e maximal normal p-subgroup of
G
G
Gi, i E I
NOTATION
522
Op(G/Opr(Gl) i n G
t h e image of
t h e general l i n e a r group of degree of p elements
n over t h e f i e l d
t h e s p e c i a l l i n e a r group of degree
n over F
P
P
=
{ g E GlgM E CGlM(N/M1
P
1
t h e n-th term of t h e lower c e n t r a l s e r i e s of
G
= < g n 1 g E G>
G
the number of double (P,Pl-cosets i n
t h e n-th term of t h e Brauer-Jennings-Zassenhaus series for G t h e n-th term of Lazard s e r i e s f o r =
G
n
(1
G
+rmn)
t h e weight of t h e F r a t t i n i subgroup of
G
n
synnnetric group of degree a l t e r n a t i n g group of degree
n
t h e group of a l l nonsingular transformations of t h e v e c t o r space TI t h e group of a l l
n x n nonsingular m a t r i c e s over R G
t h e automorphism group of
V
t h e automorphism group of an R-module =
ig l g E P
the
cE
CIICG)
p-length of
1
C
z
t h e semidirect product of t h e nilpotency c l a s s of
P
x
z
P
and
SL(2,p)
P
t h e d i h e d r a l group of o r d e r
2n, n > 2 2 n , n> 3
t h e generalized quaternion group of order t h e semidihedral group of o r d e r = 3 n-2
bP = 1, b-lab = al+P
t h e e x t r a - s p e c i a l p-group of o r d e r odd.
H
C H 2
HI
i s G c o n j u g a t e t o a subgroup of
>
p3 and exponent
HZ
p,p
NOTATION
H
= H
'G
HI
and H
523
are &conjugate
Rings, Modules, and Characters Z (Rl
the centre of R
A @ B
tensor product
charR
the characteristic of R
? Ri
direct sum
+I
direct product of rings FG
over F
the group algebra of G
the augmentation ideal of FG aug (x)
G'F
R0
the augmentation of
3:
the twisted group algebra of
G
over
F
the opposite ring of R the ring of all n x n-matrices over R the Jacobson radical of a ring R
J(V)
the radical of an R-module
SocR
the socle of a ring R
socv
the socle o f a module
V
V
the additive group of all R-homomorphisms V -
V
the n-th direct power of the composition length of the annihilator of
V
the projective cover of
I,
the unit group of a ring R the Loewy length of the Heller module of = Hom(V,F)
F
V
V
V
W
NOTATION
524
Si ( R )
t h e i - t h s o c l e of
R
si(v)
t h e i - t h s o c l e of
V
r
support of t r a c e of
z
V
t h e o u t e r t e n s o r product of
and
W
t h e inner t e n s o r product of FC-modules
U and
V
the induced module
t h e r e s t r i c t i o n of
vH lG o r 1
FG
vL AL
I d (XI
V
to
FH
t h e t r i v i a l FG-module =LQV
F = L 8 A F t h e sum of a l l two-sided i d e a l s of in X
FG
contained
V
G-conjugate of
c o n t r a g r a d i e n t module t h e intertwinning number
V
F
did F
dimension of
over
i n f (V1
i n f l a t e d module
ext(V)
extension of
XV
t h e c h a r a c t e r afforded by
V
gX
t h e c h a r a c t e r afforded by
'V
XG
t h e induced c h a r a c t e r
X+
=
V
c x fix A
t h e commutator subspace of d e f e c t group of
C E CL(G)
d e f e c t group of a block
B
t h e F-linear span of a l l 'C C E CL(C) and 6 ( C ) C_ D
with
G t h e F-linear span of a l l such t h a t D 3 6 (C)
C+ with
C E CR(C)
525
NOTATION
the F-linear span of a l l C" where C p-regular class of G with 6 ( C ) = D
t(GI
the nilpotency index of
A*
=
is a
J(FG)
Hom(A,F)
F FX
the F-linear span of X
1
=
{a E Al$(aX)
=
0
1
=
{a E AI$(Xu)
=
0
1
X
X1
01
= { a E A ~ U X= =
{a E AlXu
0 1
=
n =
{a E AlaP
for some n 2 11
IA,Al
E
the trace map
V
obstruction to the extension o f
to FG
the tensor product of representations the sum of all p-elements of KerX
=
{3:
KerT
=
Ix E FGIT(gccl
V
E FGIXv(g;c) = =
0 for all g E G I
o
the multiplicity of of w
n
R(FG1
=
FGI trtB
{a E
1
G including 1
=
for all g E
V
o
GI
as a composition factor
for all sufficiently
large n E N}
z
P'
P =
1 FS:
where S1,.
i=1
.. ,S r
are all p-regular
sections of
G.
trA
the trace
the matrix A
Kerb'
the kernel of the representation afforded by
KerB
the kernel of the block
of
B
n =
{;c
E FGlP
E IFG,FGl}
n = {z?
13:
E
XI n
= {g E
GI$
E
XI
a complement of =
J(A) in A
Z(FG) n Z(FN)
the inertia group of the block
b
V
NOTATION
526
e*
the sum of all &conjugates of
e
the relative trace map the Cartan matrix of
FG
the nilpotency index of J ( B ) the nilpotency index of J ( Z ( B ) ) the unique block corresponding to
ux ( V )
the vertex of 'I
6 (BI
the defect group of
B
the defect group of
e
xG
Cohomology Theory the group of all A-valued 2-cocycles of a coboundary the subgroup of a l l coboundaries =
Z 2 (G,A) / B 2 (G,A1
the cohomology class of obstruction cocycle = (infa)(x,yl =
a(xN,yN)
f
E . Z 2 (G,A)
G
527
Index
Action, 4, 486 faithful, 486 transitive, 5 Additive structure, 1 Algebra, 2, 52 definable over a subfield, 114 direct product of, 53 Frobenius, 123 homomorphism o f , 2 index of, 455 matrix representation of, 57 of finite representation type, 289 regular representation o f , 57 representation of, 56 separable, 114 symmetric, 123 uniserial, 285 Alperin, 392, 497 Annihilator, 27 of induced modules, 335 Anti-automorphism, 379 Asano, 496 Augmentation, 6 9 Augmentation ideal, 68 Azumaya, 36, 292 Bedi, 372 Bijective correspondence, 19 between the isomorphism classes of R and M (R)-modules, 19 betwee; the sets of ideals of R and M ( R ) , 20 bztween the a-representations and PG-modules, 145 Bilinear form, 123 associative, 123 nonsingular, 123 Bimodule, 44 Blackburn, 112 Block, 59 irreducible representation associated with, 59
nilpotent, 498 Block idempotent, 59 Brauer, 79, 81, 139, 192, 195, 200, 226, 421
Brauer correspondence, 223 Brauer homomorphism, 76 Brauer-Jennings-Zassenhaus series, 302 Brockhaus, 175, 193, 195 Broue, 498 B-subpair, 498 Burnside, 190, 225 Canonical injection, 88 Cartan invariants, 65 Cartan matrix, 65 Central character, 59,64 Centralizer, 4 Centre, 1 of a group, 4 of a ring, 1 of symmetric algebras, 133 Characteristic subgroup, 302 Characterization of defect group of a block, 2 7 1 defect of a block, 7 9 exponents of defect groups of blocks, 455
dimension subgroups, 308, 309 dimFJ(Z'(FG)) 217 elements of J(FG), 140, 177 H-injective modules, 95 H-projective modules, 9 4 J(R), 3 0 J ( V ) , 26 kernel of a block, 195 local rings, 36 p-groups with cyclic subgroups of index p , 417 p-radical groups, 357, 358, 365, 367 projective modules, 41,42 SOC V , 26 vertices, 103
.
528
INDEX
Clarke, 397, 418, 494, 495 C l i f f o r d , 141, 146, 149, 240 C l i f f o r d ’ s theorem, 141 Coboundary , 142 Cocycle, 142 Cohomology c l a s s , 143 Cohomologous cocycles, 143 C o l l i n s , 392 Commutative diagram, 40 Commutator, 301 Commutator subspace, 73 Component, 100 Composition s e r i e s , 8 equivalent, 9 factors o f , 8 length o f , 8 Conjugate of a module, 1 4 0 Conlon, 163, 167, 168 Conlon’s theorem, 163 Contragradient module, 379 Covering of a block, 229 Criteria for absence of blocks of d e f e c t zero, 372 algebra t o be Frobenius, 125 algebra t o be of f i n i t e r e p r e s e n t a t i o n type, 289 algebra t o be s e p a r a b l e , 114 algebra t o be symmetric, 126 algebra t o be u n i s e r i a l , 290 block t o be defined, 467, 471 block t o be r e g u l a r , 469 block t o be of zero d e f e c t , 180 block t o cover another block, 232 blocks t o be quasi-primary, 255 c e n t r e of a r i n g t o be indecomposable, 16 commutativity of J(FG), 260, 263, 265 commutativity of t h e p r i n c i p a l block, 257 complete r e d u c i b i l i t y of a module, 363 complete r e d u c i b i l i t y o f induced modules, 87 dim#(B) = dimFB(l-l/u), 500 dimpJ(FG) = dimJ(FG1 =
, 193 IGl (PI - 1. 198
each block t o be quasi-primary, 255 element t o be i n r a d i c a l , 2 7 , 30 equivalence of composition s e r i a s , 9 e x i s t e n c e of normal p-complement, 225 e x i s t e n c e of p r o j e c t i v e cover, 48 e x t e n d i b i l i t y of modules, 146 ( e , f ) t o be simple induction p a i r , 336 ( e ,f)t o be simple r e s t r i c t i o n p a i r ,
32# ,“wc
0, 338 t o be completely r e d u c i b l e , 341
( e p ) H t o be completely r e d u c i b l e , 344
eJ(FH)FG C FG*J(FH), 344 eJ(FH)FG FG*J(FH)e, 344 f a c t o r algebra t o be Frobenius, 130 f a c t o r algebra t o be symmetric, 130 f i e l d t o be p e r f e c t , 114 f i e l d t o be a s p l i t t i n g f i e l d , 55,56 Frobenius algebra t o be u n i s e r i a l , 291 FG t o be of f i n i t e r e p r e s e n t a t i o n type, 298 FG t o be l o c a l , 106 PG t o be l o c a l , 152 fW t o be completely reducible, 341
(g)G
t o be completely r e d u c i b l e ,
344H
fJ(FG) = fJ(FH)FG, 345 FG J(FH)M 2 J(FG)M, 346 FG t o be indecomposable, 1 1 2 group a l g e b r a t o be semisimple, 7 1 group t o be d e f e c t group, 278 group t o be Frobenius, 184 group t o be of p-length 1, 416, 426 group to be p - n i l p o t e n t , 244, 245 group t o be 2 - n i l p o t e n t , 248 group t o be p - r a d i c a l , 361 group t o be r e g u l a r , 418 group t o s p l i t over a normal subgroup, 417 homomorphism t o be e s s e n t i a l , 46 H t o be of p’-index, 352 H2(G,F*) t o be i n f i n i t e , 150 i d e a l t o be i n J ( R 1 , 3 1 i d e a l t o be Jacobson r a d i c a l , 29 i d e a l t o be n i l p o t e n t , 56 induced module t o be indecomposable, 168 induced module t o be p r o j e c t i v e , 99 i r r e d u c i b l e module t o be H-projective 352 isomorphism of p r o j e c t i v e modules, 49 isomorphism of t w i s t e d and ordinary group a l g e b r a s , 1 4 4 , 151
J(B)
J(B1
J ( F N ) B , 281 = B J ( Z ( B ) ) , 246 =
J ( B ) = I ( D ) * B , 282 J ( R ) t o be n i l p o t e n t , 31, 34 J ( A ) t o be p r i n c i p a l i d e a l , 292 J(FG)V = J ( F N ) V , 281 J(FG) t o be p r i n c i p a l i d e a l , 299 J(FG) = R(c), 175 J(FG) C FG*J(FH), 338 J(FG)’-= 0 , 250 J ( F G ) C Z(FG), 251 J(FG) FG*J(Z(FG),246 J(FG) C J(FH)FG, 338 J(FH) J ( F G ) , 338 J(FG)f-= FG*J(FH)f, 345 J(FG)e = FG.J(FH)e, 347 J ( F G ) e = J ( F H ) F G e , 347
INDEX
529
J ( F G 1 = FG.J(FH), 352 t ( G ) < (a+ ) (p 1) + 1, 447 J ( F G ) = JLFHIFG, 352 449 t(G) = pa-', J ( B 1 2 = 0 , 249 t ( G ) = U(p-1) + 1, 419, 427, 447 J ( F G 1 3 = 0 , 420 t(G) = pa, 419 J ( V ) = 0 , 25 t isted group algebra to be local, 152 J ( Z ( F G ) ) dto be an ideal, 495 to be completely reducible, 338 # 0 , 498 Z, E J ( V ) , 27 J(Z(FGI)P L(P(lG)I = 3, 494 Curtis, 79, 188, 209 lifting idempotents, 156 module to be absolutely irreducible,55 Dade, 154 module to be artinian, 5, 7 Defect, 77 module to be completely reducible, 12 of a block, 77 module to be finite direct sum of of a class, 77 irreducible modules, 23 Defect group, 77 module to be finitely cogenerated, 5 of a block, 77 module to be finitely generated, 6 of a block idempotent, 77 module to be flat, 46 of a class, 77 of covering blocks, 466 module to be free, 161 module to be injective, 43, 285 Dimension subgroups, 302 module to be injective hull, 288 Direct decomposition of rings, 15 module to be irreducible, 28 Direct power of a module, 19 module to be H-injective, 96 Direct product of p-radical groups, 360 module to be H-projective, 96 Division ring, 21 module to be noetherian, 6,7 module to be projective, 41, 42 Eilenberg , 126 module to be projective cover, 51 Equivalent representations, 57 modules to belong to the same block, 64Erdmann, 421, 494 modules to be linked, 63 Essential homomorphism, 46 module to be separable, 114 Essential submodule, 287 module to have composition series, 8 Exact sequence, 40 module to have finite decomposition, 7 Extension of ground field, 52 module to have maximal submodule, 6 External direct sum, 4 module to have zero radical, 25 to be completely reducible, 338 Feit, 493 Mu nilpotency of J ( R ) , 31 Fermat prime, 446 P ( l 1 = FG, 377 Filtration of 1 ( G ) , 304 G p-regularity, 196 Fixed-point-space, 186 ring to be full matrix ring, 18 Fong, 171, 474 ring to be indecomposable, 16 Fong correspondence, 474 ring to be local, 36 Fong correspondent, 474 ring to be simple artinian, 32 Fong's dimension formula, 170 ring to have finite decomposition, 17 Formanek , 177 Soc FG = FGc, 176 Frattini subgroup, 320 Frobenius, 183 submodule to be fully invariant, 14 Frobenius complement, 183 submodule to be radical, 25 submodule to be socle, 25 Frobenius kernel, 183 t ( B ) = pa-l+p-1, 442 Fully invariant submodule, 14 t C B ) > pn-' 4 - 1 , 457 Glauberman, 416, 417 t ( B ) = pd 490 Gorenstein, 183, 416, 417, 418, 421 t B pdL1+p-l, 491 Gow, 226 t C B ) < pd, 491 Green, 168, 218, 277 t ( B ) = /DI,492 t ( B ) = 3 , 493 Group, 4 t ( G ) = 4, 499 alternating, 5 composition series of 303 t ( G ) = dim#(FG) +1, 421 cyclic, 5 t ( G ) = t ( P 1 , 418 dihedral, 320 t ( G 1 = pa- +p-1, 442 elementary abelian, 303 pa- < t ( G ) < p a , 442 extra-special, 320 t ( G ) = (ail)Cp-1) + 1, 447 Frobenius, 183
?
&'
530
g e n e r a l l i n e a r , 413 generalized quaternion, 320 involved, 416 metacyclic, 318 multiplicative, 4 m u l t i p l i c a t i v e of a f i e l d , 4
M(p), 320
M ( p ) , 320 nylpotent, 301 nilpotency c l a s s o f , 301
PSL(2,q), 421 p-constrained, 112 p-element o f , 4 p '-element o f , 4 p-length o f , 283 p - n i l p o t e n t , 197 p-radical, 357 p-regular c l a s s o f , 4 p-regular element o f , 4 p-regular s e c t i o n o f , 138 p-singular c l a s s of, 4 p-solvable, 110 Q d ( p ) , 416 r e g u l a r , 418 semidihedral, 320 s o l v a b l e , 110 s p e c i a l l i n e a r , 413 symmetric, 5
S(n), 417 Group a l g e b r a , 67 augmentation i d e a l o f , 68 c e n t r e o f , 71 H a l l , 151 P. H a l l , 414, 415, 417 Hall-Higman's lemma, 415 Hall-Higman's theorem, 417 Hamernik, 255, 257 Height, 304 Heller module, 376 G. Higman, 415, 417 D. Higman, 94, 96, 225, 298 H i l l , 312 Hochschild, 225 Homogeneous component, 14 Homomorphism of a l g e b r a s , 2 Huppert, 112, 321, 416, 417, 418 M.
Ideal, 1 nil, 1 nilpotent, 1 Idempotent, 1 primitive, 1 c e n t r a l l y primitive, 1 orthogonal, 1 I i z u k a , 178 Inductive set, 3 I n e r t i a group, 141, 229 I n f l a t i o n map, 146
INDEX
I n j e c t i v e h u l l , 287 I n t e r n a l d i r e c t sum, 3 Intertwining number, 90, 363 I t e r a t e d s o c l e , 378 Jacobson, 114 Jacobson r a d i c a l , 22 dimension o f , 66 of d i r e c t sums, 25 of d i r e c t products, 31, 119 of f u l l matrix r i n g s , 31 of module, 22 of r i n g , 22 of submodule, 24 Jennings, 307, 308, 309, 311 Jennings' formula, 311, 321, 322, 3 3 0 Jennings' theorem, 307 Jordan-Hb;lder theorem, 9 Karpilovsky, 151, 423 Kasch, 298 Kawada, 226 Kernel of a block, 195 Kernel of a module, 195 K h a t r i , 343, 359, 360 Kneser, 298 Knorr, 277, 281, 335, 336, 343, 466 Koshitani, 240, 262, 318, 321, 325, 329,
416, 417, 419, 427, 440, 442, 490, 491, 492, 494, 495, 498, 499, 500 Krull-Schmidt theorem, 39 Kiilshammer, 134, 178, 207, 214, 246, 453, 455, 456, 459, 474, 484, 487, 495 Kiilshammer's s t r u c t u r e theorem, 484 Kupisch, 298 Lattice, 3 isomorphism, 3 homomorphism, 3 of submodules, 3 Lazard, 310 Lazard s e r i e s , 302 L e f t a n n i h i l a t o r , 127 Left transversal, 4 Lie product, 74 Linear equivalence, 143 Loewy l e n g t h , 284, 375 of induced modules, 400 of p r o j e c t i v e covers, 391, 407 Loewy s e r i e s , 284 Loncour, 119 Lorenz, 389, 393, 394, 407, 409, 425 Lower c e n t r a l s e r i e s , 301 Mackey decomposition, Malcev, 209 Maschke's theorem, 7 1 Matrix u n i t s , 17 Maximal element, 3 Maximal submodule, 3
lo1
INDEX
531
Michler, 195, 226, 235 Nilpotent element, 1 Ninomiya, 265, 266, 323, 329, 343, 357, Modular law, 3 Module, 2 360, 362, 372, 419, 422, 425, 494, absolutely irreducible, 54 499 artinian, 5 Noether-Skolem Theorem, 241 Normal closure, 348 basis of, 4 Normalizer, 4 completely reducible, 12 Normal p-complement, 197 composition length of, 10 direct product of, 3 N -sequence, 302 dual, 284 N!?mber of irreducible modules, 81 external direct sum of, 3 faithful, 28 Okuyama, 175, 372, 462, 493, 498 Orbit, 4 finitely cogenerated, 5 Outer tensor product, 116 finitely generated, 5 flat, 45 Partially ordered set, 3 free, 4 Passman, 108, 110, 310, 418, 423, 464 G-invariant, 141 p-complement, 107 head of, 90 Perfect field, 113 homogeneous, 14 Primitive ideal, 28 H-injective, 91 Principal block, 195 H-projective, 91 Projective cover, 46 imprimitive, 82 Projective matrix representation, 143 indecomposable, 3 Projective representation, 143 induced, 82 inflated, 149 completely reducible, 144 injective, 42 irreducible, 143 in a block, 61 a-representation, 143 Proper submodule, 3 irreducible, 2 Puig, 498 internal direct sum of, 3 linked, 63 Quasi-primary block, 253 left, 2 noetherian, 5 Regular block, 469 of finite length, 10 Reiner, 79, 188, 209, 241 principal indecomposable, 12 Relations between projective, 41 radical of, 22 Jacobson radical of a module and a reducible, 2 submodule, 24 Socle of a module and a submodule, 24 regular, 123 Relative trace map, 268 separable, 113 socle of, 22 Representation, 56 S-projective, 274 completely reducible, 57 uniserial, 285 faithful, 72 indecomposable, 57 unital, 2 Modular law, 3 irreducible, 57 Miller, 500 of algebras, 56 Morita, 105, 228, 238, 240, 255, 284, of groups, 72 292, 299, 419, 424, regular, 57 Morita's theorem, 240 underlying module of, 56 Motose, 119, 189, 265, 266, 315, 318,323,Restrictionof a module, 82 325, 326, 328, 329, 343, 351, 360, 362,Hetraction, 275 372, 419, 422, 425, 427, 428, 429, 43O,Reynolds, 179, 474 437, 440, 444, 445, 446, 447, 449, 495,Right annihilator, 127 496, 497, 499, 5 0 0 Right transversal, 4 Ring, 1 Nakayama, 126, 130, 240, 290, 292 associative, 1 Nakayama's lemma, 29 commutative, 1 Nakayama reciprocity, 90, 367 direct product of, 2 Nesbitt, 192, 200, 226, 421 homomorphism, 1 Nilpotency index, 118 identity element of, 1 Nilpotent conjugacy class, 218 indecomposable, 16
532
INDEX
Upper p-series, 282 local, 36 Upper bound for t(Z( B ) ) , 462 of endomorphisms, 2 Unit (left, right), 3 0 of endomorphisms of a completely Universal characterization of induced reducible module, 15 modules, 88 of endomorphisms of direct powers, 19 of matrices, 17 Vertex, 100 opposite, 1 Villamayor, 108 semisimple, 22 socle of, 22 Robinson, 214 Wallace, 189, 192, 193, 198, 250, 251, 260, 313, 392, 418, 419, 421 Ward, 154 Saksonov, 362, 365, 368, 369, 372 Watanabe, 178 Scalar matrix, 17 Wedderburn, 209 Schanuel's lemma, 376 Wedderburn-Malcev theorem, 209 Schur's lemma, 21 Second cohomology group, 142 Weight, 304 semidirect product, 486 Willems, 108, 154, 168, 391 Separable element, 113 Zassenhaus, 190 Separable extension, 113 Zorn's lemma, 3 Separable polynomial, 113 Short exact sequence, 40 split, 40 Sibley, 392 Simple induction pair, 336 Simple restriction pair, 336 Shoda, 240 Snider, 177 Socle, 22 of Frobenius algebras, 131, 134 of module 22, Source, 101 SDieael. 250. 251 Splitting field, 55 Splitting homomorphism, 40 Stabilizer, 4 Subgroup, 4 generated by a set, 4 Submodule, 14 essential, 26 fully invariant, 14 superfluous, 26 Subring, 1 Support, 68 of central idempotents, 76 Swan, 154 System of imprimitivity, 82 Tensor product, 44 inner, 97 of algebras, 117 of Frobenius algebras, 126 of modules, 44 of projective representations, 149 Thompson, 183 Trace map, 137 Transitivity of induction, 87 Tsushima, 139, 177, 246, 361, 363, 366, 367, 371, 372, 418, 419, 499 Twisted group algebra, 144
