OKUYAMA, TETSURO

TSUSHIMA, YUKIO

Osaka City University Repository

Okuyama, T. and Tsushima, Y.
Osaka J. Math.
24 (1987), 391-394
ON THE THEOREMS OF GASHUTZ AND WILLEMS
Dedicated to Professor H. Tominaga on his 60th birthday
TETSURO OKUYAMA AND YUKIO TSUSHIMA
(Received Fabruary 13, 1986)
1. Introduction
Let p be a prime and F=Z/(p). Let G be a finite group. By a p-chief
factor, we mean a chief factor group V=HIK which is a^>-group, where HZDK
are normal subgroups of G. Since V is elementary, it is regarded as an ir-
reducible right FG-module. If V has a complement in G/K, then it is called
complemented. Now let us fix a chief series of G;
E: 1 = Q ^
 n
For an irreducible .FG-module U we put
m(E, U) = I {i; GilGi^—U and <?,/<?,_! is complemented.} |
Let / be the radical of FG and let e be a primitive idempotent of FG such that
eFGjeJ is isomorphic to the trivial .FG-module F. Recently it is shown that
Theorem 1 (Willems [3]). If V is a complemented p-chief factor of G,
then it appears as a component of ej/ej2 with multiplicity at least m(E, V).
On the other hand the following result is known as a theorem of Gashutz
(see [1] Theorem 15.5 and Remark 15.6).
Theorem 2. Suppose that G is p-solvable. Then if U is a simple com-
ponent of ejjej2, U must be isomorphic to a complemented p-chief factor of G and
it appears exactly m(E, U) times in ejjej2.
All known proofs of Gashtitz's theorem involve essentially cohomological
arguments and take some efforts to understand. In this short note we shall give
an elementary approach to both theorems, which will be very lucid as well.
2. Preliminary Lemmas
In this section we prove several Lemmas. Some of them are possibly
known, but we give proofs for the completeness. The notation used in the
introduction will be fixed throughout. In addition we use I(G) to denote
392 T. OKUYAMA AND Y. TSUSHIMA
the augmentation ideal of FG. Note that eI(G)=eJ.
Lemma 1. Let Gt>H. Then there exists an FG-epimorphism
a: I(H)II(Hf -> eI(H)FGjeJI(H),
where I{H)jI{H)2 is considered as an FG-module by conjugation.
Proof. Define /3: I(H) -* eI(H) FG/eJI(H) by /3(h-l)=e(h-l) + eJI(H)
for Aeff. We observe that for heH andgGG
(1) e(h-iy=e(h-l)g + e(l-g)(¥-l) and e(l-g)(h*-l)tEeJI(H),
(2) eg(l-h)=e(l-h) + e(g-l)(l-h).
The first equality implies that ft is an i^G-homomorphism, while the second
one implies that /3 is an epimorphism since I(H)FG=FGI(H). Since Ker
/3Z)I(H)2, /? yields a as desired.
L e m m a 2. Let V be a minimal normal p-subgroup of G which is comple-
mented. Then we have the following:
(1) For any primitive idempotent f of FG, fl( V)FG <£ fj2y
(2) eI(V)FG+
Proof. Let T be a complement of V in G; G=TV, TPiV=l. We
have that FG=FT 0 I(V)FT and J=J0 © I(V)FT, where / 0 denotes the radical
of FT. Nothing changes in the above statements if we replace / and e with
their conjugates by units of FG. In particular we may assume that/ , e^FT
by the lifting idempotent theorem since I(V)FTdJ (see Landrock [2] Theorem
1.5).
We have J2=I{V)2FT + I(V)J0 + J0I(V) + Jl Note that I(V)FT=
FTI(V)Z)J0I(V) and I(V)FTnjo=O. So that i£fI(V)FG=fI(V)FTcfJ2, we
get the following equality from the above:
fI{V) FT =fI{VfFT+fI{V)JQ +fjo I(V).
Let t be any non-zero element in the socle of FTf By multiplying both sides
by ty we get tI(V) FT=tI(V)2FT + tI(V)/0, whence it follows that tI(V)FT
=tI(V)2 FT by Nakayama's Lemma. Furthermore since both of them are
.FF-modules, we get tFTI(V)=0 by Nakayama's Lemma again. This is im-
possible and hence//(F) FG<tfJ2.
To show the second, recall that V^I(V)/I(V)2 (see Willems [3]). This,
together with Lemma 1, yields an FG-epimorphism
cp: V -> eI(V) FG/eJI(V) -> eI(V) FG + e]2\ej2.
Since V is irreducible and eI(V)FG + ej2jej2 is not zero from the above, <p must
be an isomorphism. This completes the proof of Lemma 2.
THEOREMS OF GASHOTZ AND WILLEMS 393
Lemma 3. Let V be a minimal normal p-subgroup of G. If eI(V) FGcteJ2,
then V is complemented.
Proof. There exists an .FG-submodule L of ej such that
ejjej2 = L/eJ2 © eI(V) FG + efjej2.
Let T={xE:G; e(x—1)GL}. We want to show that T is a complement of
V. It is clear that T is a subgroup of G, since e(xy— l)=e(x— l)y + e(y— 1)
for any x, y^G. It is also clear that T1&V from the assumption. So it re-
mains only to show that G=TV. Note that FG=FV + S (x—l)FV, where
x runs through a set of coset representatives of V in G. It follows from this
that eI(V) FG=eI(V) + eJI(V). Therefore we have eJ=eI(V) FG + L=
eI(V) + L. Let x(=G. Then e(x—l)=ea + b, where a&I(V) and b(=L.
Since the isomorphism F—/(V)jl{Vf is given by v h-> (v— 1) + /(I7)2 for ^ G F ,
we may write #=(^—1) + u with & G F and u^I(V)2. Thus e(x—v)^L and
we get ^ ^ - I J G L . This implies that xv~1^. T and so G=TV, as asserted.
Lemma 4. Suppose that G is p-solvable and let HZ)K be normal subgroups
of G. If H\K is a p'-group, then eI(H) FG=eI(K) FG.
Proof. If we put G=G/K, then I(H) FG/I(K) FG^I(H) FG. So it is
sufficient to show that eI(H) FG=0 by assuming K=l. Let M be a ^-com-
plement in G. Then we can take e as e= 2 #. Since MIDH, we get £/(#)
=0, as asserted. ' '*eJr
3. Proof of the theorems
Using the notation in the introduction, we have an ascending series of
right ideals;
0 c - c J ( G M ) FG + efdeliGi) FG + eJ2(Z-~ deI(Gn) FG = eFG .
If GijGi-x is a complemented p-chiei factor of G=G/Gi-ly then we have by
Lemma 2
eI(G{) FG + eJ2leI{G^) FG + eJ*~eI{Gi) FG + epIep^G./G^ .
So the Theorem 1 is obvious, while the Theorem 2 is immediate from this and
Lemmas 3 and 4.
References
[1] Huppert B. and Blackburn N.: Finite groups II, Springer-Verlag, Berlin/Heidel-
berg/New York, 1982.
394 T. OKUYAMA AND Y. TSUSHIMA
[2] Landrock, P.: Finite group algebras and their modules, Cambridge University
Press, London/New York, 1983.
[3] Willems, W.: On p-chief factors of finite groups, Comm. Algebra 13(1985), 2433-
2447.
Department of Mathematics
Osaka City University
558 Osaka, Japan


On the theorems of Gashutz and Willems

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On the theorems of Gashutz and Willems

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