OKUYAMA, TETSURO

Okuyama, Tetsuro

Osaka University Knowledge Archive

p-radical groups are p-solvable

Osaka City University Repository

Okuyama, T.
Osaka J. Math.
23 (1986), 467-469
p-RADICAL GROUPS ARE p-SOLVABLE
Dedicated to Professor Hirosi Nagao for his 60th birthday
TETSURO OKUYAMA
(Received January 24, 1985)
Let k be an algebraically closed field of characteristic p>0. Let G be a
finite group with Sylow ^ -subgroup P. Following Motose and Ninomiya [3] we
call G />-radical if kP
G
 is completely reducible, where kP is the trivial &P-module.
Our aim in this paper is to prove the following theorems.
Theorem 1. If G is p-radical, then G is p-solvable.
Theorem 2. Let G be a p-radical group with Sylow p-subgroup P. Then
the following hold;
(1) If D=P f]Px for some x in G, then D is a vertex of some simple kG-
module.
(2) // D=P Π Px for some x in CG{D)y then D is a defect group of some p-
blockofG.
We will write V \ W if a AG-module V is isomorphic to a direct summand
of a ΛG-module W. For &G-modules V and W and a subgroup H of Gy let
(F, W)G=HomkG(Vy W) and (V, W)^=TH>G(V, W)Hy where THtG is the trace
map from (F, W)H to (F, Wf.
1. Preliminaries
Throughout this paper we let G be a ^-radical group with Sylow ^-sub-
group P and put Y=kP
G
. In this section we shall prove two lemmas which
will be used to prove the theorems stated in the introduction.
Lemma l If S is a simple kG-module with vertex Q, then every indecom-
posable direct summand of SP is isomorphic tr k/ for some AdP which is conjugate
toQ.
Proof. Since Y is completely reducible, (Y, S)G=(Yy S)§ and (F, S)£ = 0
if R does not contain any conjugate of Q. Let X be an indecomposable direct
summand of SP. Then by Mackey decomposition theorem X^kA
p
 for some
AdP such that A is contained in some conjugate of Q. By the isomorphism
468 T. OKUYAMA
Hornby, S)^(Homk(kP, SP))G we have TGtA((Y, S)^)φO (see Lemma 3.5, II in
[1]). Thus by the above remark A contains some conjugate of Q and there-
fore A is conjugate to Q.
Lemma 2. Let Q be a subgroup of P and put N=NG{Q) and R=P Π N.
Let V be an indecomposable direct summand of kRN. If VR has an indecomposable
direct summand with vertex Q, then V also has Q as a vertex.
Proof. For our proof of the lemma Scott's study in [5] is very useful.
Let Ω be the set of right cosets of P in G. Then Y is the permutation module
kΩ. If ΩQ is the set of fixed points of Q in Ω, then X=kΩQ is a &iV-module,
I = Θ Σ ^ n ί where x ranges over the set of representatives of (P, iV)-double
cosets in G with PXZ)Q and kRN\X. Note that every indecomposable direct
summand of X has a vertex containing Q as Q<\N. By Scott's investigation in
(section 3, [5]) there exists a ^-algebra homomorphism/ from (Yy Y)G to (X, X)N.
The map / induces the epimorphism from (Y, Y)ρ to (X, X)% (see Proof of
Theorem 3(b), [5]). Since (Y, Y)G is semisimple we conclude that (X, X)Q is
an ideal of (X,X)N with J((X9X)N)Γi(X,X)%=09 where J((X, X)N) denotes
the Jacobson radical of (X> X)N. Then it follows that (X} X)Q is a direct
summand of (X, X)N as algebras. Let Fbe an indecomposable direct summand
of kR
N
 and assume that VR has an indecomposable direct summand with vertex
Q. Then kQR is a direct summand of VR by Mackey decomposition theorem and
therefore (F, kRN)% Φ 0. Thus (V, X)Q Φ 0 and this is implies that an idempotent
in (X, X)N corresponding to V is in (X, X)% as (Xy X)Q is an algebra direct
summand of (Xy X)N. So V also has a vertex Q and the result follows.
2. Proof of Theorem 1
Let S be a simple ΛG-module in the principal ^>-block of G and Q be its
vertex. By Lemma 1 SP= Φ Σ kQχ
p
 for some x's with QxdP. By the result of
Knϋrr (Corollary 3.6, [2]) Q is a defect group of the principals-block of QCG(Q)
and therefore is a Sylow ^-subgroup of QCG(Q). Thus Z(P)czQx for every x
with QXCLP. SO Ker S ^Z(P) and it follows that Op,p(G)θ>Z(P). As G/O^G)
is also ^-radical, the theorem follows by induction on the order of G.
3. Proof of Theorem 2
First we shall prove the statement (1) in Theorem 2. For any Sylow in-
tersection D=PX Π P, kD
p
 is a direct summand of YP by Mackey decomposition
theorem and therefore is a direct summand of SP for some simple &G-module
S. Then the result follows from Lemma 1.
Next we shall show the statement (2) in Theorem 2. Let D=Pxf)P
where x is in CG(D) and put N=NG(D), H=DCG(D) and R=Pf)N. Since x
is in CG(D) and D=RX f] R it follows that kDR \ kRRHR. Then by Lemma 2 kRRH
^-RADICZAL GROUPS ARE />-SOLVABLE 469
has an indecomposable direct summand V with vertex D as kR
RH
 \ kR
N
RH. Let
W be an indecomposable Aiϊ-module such that WRH= V and let b be a ^ >-block
of H which contains W. We claim that WN is completely reducible. By the
result of Scott (Theoremc, [5]) every indecomposable direct summand of kRN
with vertex D is the Green correspondent of an indecomposable direct summand
of Y and is therefore a simple ΛΛΓ-module by (Lemma 2.2, [4]). Since
WN\ VN\{kRRH)N—kRN and every indecomposable direct summand of WN has a
vertex D we have that WN is completely reducible by the above remark and
our claim follows. Since WN is completely reducible, W is simple and is a
unique simple &£/-module in b as H=DCG(D). Let B be a ^-block of iV
which covers i. Then every simple &/V-module in B is a submodule of WN
and therefore direct summand of WN. This implies that B has a defect group
D. So G has a ^-block with defect group D by Brauer's First Main Theorem.
References
[1] W. Feit: The representation theory of finite groups, North-Holland, Amster-
dam, New York, Oxford, 1982.
[2] R. Knδrr: On the vertices of irreducible modules, Ann. of Math. (2) 110 (1979),
487-499.
[3] K. Motose and Y. Ninomiya: On the subgroups H of a group G such that J(KH)
KGz>J{KG), Math. J. Okayama Univ. 17 (1975), 171-176.
[4] T. Okuyama: Module correspondence in finite groups, Hokkaido Math. J. 10
(1981), 299-318.
[5] L.L. Scott: Modular permutation representations, Trans. Amer. Math. Soc. 175
(1973), 101-121.
Department of Mathematics
Osaka City University
Osaka 558, Japan



p -radical groups are p -solvable

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p -radical groups are p -solvable

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