TSUSHIMA, YUKIO

Tsushima, Yukio

Osaka University Knowledge Archive

A note on Cartan integers for p-solvable groups

Osaka City University Repository

Tsushima, Y.
Osaka J. Math.
20 (1983), 675-679
A NOTE ON CARTAN INTEGERS FOR
p-SOLVABLE GROUPS
YUKIO TSUSHIMA
(Received September 10, 1981)
1. Introduction
Let B be a p-block of a finite group G. As is well-known, if the Cartan
integer c
φφ
=l for some φ^IEτ(B), then B must be a block of defect zero. On
the other hand there are various blocks in which the second smallest case, name-
ly c
φφ
=2 for some φ occurs, though they do not seem generally to have specific
natures in common. However in such blocks of ^-solvable groups we can
show the following, which is the purpose of this paper.
Theorem. Let G be a p-solvάble group and B a p-block (ideal) of R[G]
with defect group D. If the Cartan integer c
φφ
—2 for some φ^IBr(B), there
exists a group T which is involved in G and satisfies:
T has a normal Sylow p-subgroup Q isomorphίc to D and if H is a p-com-
plement of T, then H acts faithfully and transitively on Q*. B is ίsomorphic
to the full matrix ring M(n, R[T]) over R[T] of degree n=deg φ as R-algebras.
In particular D is elementary and \H \ = (\D\ — 1) (kB—1B), where kB—\Irr(B)\
andlB=\IBr(B}\.
Here "T is involved in G" means that T is isomorphic to a homomorphic
image of a subgroup of G and £)* is the set of non-identy elements of Q. Note
that the above T has a double coset decomposition T—H \jHgH (g^Q)9 so
it can be represented as a (^-solvable) doubly transitive permutation group
and it holds that c
φφ
=^2 for every linear character φ of T. Such permutation
groups were classified by Huppert [3] and Passman [5] and as a matter of fact
the result will take an essential role in the proof of the above Theorem.
NOTATION. G will denote a finite group and p a fixed prime integer.
We fix a />-modular system (L, R, F), namely I? is a valuation ring of rank one
with quotient field L of characteristic zero and residue field F of character-
istic p. We assume that L contains a primitive |G|-th root of unity. All
modular representations will be considered over F and by a p-block of G we
mean a block ideal of the group ring R[G]. As usual Irr(β) and IBr(.B) denote
the sets of irreducible L-characters and irreducible Brauer characters of B
676 Y. TSUSHIMA
respectively. Finally for a positive integer n, np and n' denote the ^>-part and
the p '-part of n respectively.
The author would like to thank Professor G. Michler for his preprint
[4] which motivated this work and also Professors K. lizuka, T. Okuyama
and A. Watanabe for their valuable advice.
2. Preliminary lemmas
First of all we mention the following which is a direct consequence of
the classifying theorems of ^-solvable doubly transitive permutation groups
due to Huppert [3] and Passman [5].
Lemma 1. Let (G, Ω) be a p-solvable doubly transitive permutation group
with non-trivial normal p-subgroup. Then the stabilizer G
a>b of a, ieΩ(αφi)
has a normal Sylow p-subgroup and its complement is cyclic.
Proof. We may assume that G is the semidirect product G~NQ^ N (Ί Q
= 1, in which Q is a minimal normal jf)-subgroup of G of order \Ω\=pn and
N acts transitively on <g*. So G
ab=Nx=CN(x) for some x^Q*. In case of
' G
"semilinear transformations", Q is identified with the Galois field GF(pn) and
then N
x
d the Galois group of GF(pn), which is cyclic (Take x from the prime
G
field). In exceptional cases, we have \Q\=p2 or 34. If \Q\-p2, then Nd
GL(2, p) and our assertion is obvious (consider the stabilizer in GL(2, p) of
the vector (1, 0)e(Z/(p))2). If |g|^34, then "case by case" arguments prove
easily our assertion. For example, the cyclic group of order eight generated
/ 1 0 1 O v
by ~~0 is the stabilizer of the vector (0 1 0 — l)e(Z/(/>))4 in the
\ 0-1 0 0'
first one of the three groups listed on pp. 127 of Huppert [3].
The next Lemma was essentially noted by Brauer and Nesbitt [1].
Lemma 2. The Carton integers c
φφ
 are same for all the linear characters
φ of G. If G has a p-complement, say H, then this common integer is equal to
the number of (H, H)-cosets of G.
Proof. Let η be the principal indecomposable Brauer character corre-
sponding to the trivial character 1G of G. If φ is linear, then 9^97 is the principal
indecomposable Brauer character corresponding to φ and hence we have c
φφ
=
(φη, φη)=(η, η)=c
n
. In case G has a ^ -complement Hy we have η—G®HlH
and so c
u
 is equal to the number of (Hy ί/)-cosets of G by Mackey decomposi-
tion.
CARTAN INTEGERS FOR ^-SOLVABLE GROUPS 677
Lemma 3. Suppose that there exists φ^IBr(G) such that c
φφ
=2. If
OP(G) is not trivial, then it is a unique normal p-subgroup and φ belongs to a
block of G/OP(G) of defect zero.
Proof. Let Q be any non-trivial normal ^ -subgroup of G and let / denote
the (nilpotent) ideal of A=F[G] generated by {#—1; x^Q}. Let e be a
primitive idempotent of A such that φ is afforded by the socle of Ae. Suppose
that Ae/Ie is reducible. Since it is a principal indecomposable F[G/ζ)]-module,
we have c
φφ
>2 (as F[G/Q] is a symmetric algebra). On the other hand we
see that le is not zero and contains the socle of Ae. In fact it is easy to see
that the right annihilator ideal of / in A is the principal ideal σA with σ— V!#
(x^Q), which is square zero and contains no idempotent. Combining with
the above, we get c
φφ
>3, contradicting the assumption. Therefore Ae/Ie is
irreducible and φ belongs to a block of G/Q of defect zero. In particular it
follows that φ(l)p=^[G: Q]p. Since Q is arbitary, we have | Op(G) \ = \Q\, or
Op(G)=Qy completing the proof of Lemma 3.
Lemma 4. Let G be a p-solvable group and assume that Op'(G) is central.
If there exists φ^IBr(G) such that c
φφ
=2, then φ must be linear. In particular,
G has a normal Sylow p-subgroup S and if H is a p-complementy it acts on S*
transitively.
Proof. By the assumption and Lemma 3, Q=Op(G) is a (non-trivial)
minimal normal subgroup. Hence it has a complement in G, that is, if we
let N be the normalizer in G of a ^-complement of O^/(G), we get G=NQ
by Frattini argument and NΓ\Q=1 by the minimality of Q. By Lemma 3,
φ belongs to a block of N of defect zero, so it can be regarded as a principal
indecomposable Brauer character of N and then has the form φ=N®ff\ for
some λ^Irr(/f) by Fong[2], where H is a ^-complement of N (and necessarily
of G). Put η~G®Nφ=G®H\. We claim that 77 is the principal indecomposa-
ble Brauer character of G corresponding to φ. In fact i f/ is a primitive idem-
potent of F[H] such that F[H]f affords the Brauer character λ, then using the
same notation as in the proof of Lemma 3 we see that F[G]f/If^F[G]/I®H
F[H]f^F[N]®HF[H]f affords the Brauer character φ. Therefore cφφ=(ηy η)=
®Nφ) by Mackey decomposition, where g runs through a set of repre-
sentatives of (N, 7V)-cosets of G and the inner product (φ>g®Nφ) is taken over
gNg^ΓiN. Since G=NQ=NQN, any (N, 7V)-coset is represented by an
element of Q. Moreover if g^Qy thengNg~1ΠN=CN(g) and φ=g®Nφ on it.
So c
φφ
=2 forces that G=N \J NgN and l+(φ, φ)Cj!f(g)=2 for (all) g<=Ξ Q*. This
means of course that N acts transitively on Q*, having O^/(G) as its kernel
and φcχ(gϊ is irreducible. On the other hand noting that O^/(G) is central,
we see from Lemma 1 that CN(g) has a normal Sylow ^-subgroup and its com-
678 Y. TSUSHIMA
plement is abelian. Combining the aboves, we conclude that (p,\CN(g)\)=\
and φ is linear. In particular we have G=H(jHgH,g^Q by Lamme 2 and
it follows from this that Q is a Sylow ^-subgroup of G. This completes the
proof of Lemma 4.
3. Proof of the Theorem
Let K=Op'(G). If B is a principal block, we may assume that K=ί.
Then T=G satisfies the conclusion of the Theorem by Lemma 4. In general
there exists ξ^lΐr(K) such that Irr(.B)cIrr(G|^). If / is the inertia group
of ξ in G, there exists a p-block Bf of / which has the same Cartan matrix as
that of B and B^M(m, B
r
) for some m>\ (Fong [2], Tsushima [7], more nota-
bly B is isomorphic to the corestriction algebra CoresfBf or the induced alge-
bra IndfBj in the languages introduced by M. Broue and L. Puig). So if
G>/, we get our assertion by the induction on the order of G. Therefore
we may assume that G=I. Following Fong, we consider a central extension
, where Z is a cyclic group of order prime to p. Moreover from the con-
struction we see that Zc[G* G*] (Reynolds [6]). There exists a ^-block 5*
of G* which has the same Cartan matrix as that of B and B^M(m, £*) for
some m>l (Fong [2], Tsushima [7]). If c
φφ
=2 for some <p<ΞlBr(J3*), then
φ must be linear by Lemma 4 and hence ker φdZ=Op'(G*). Therefore B*
must be the principal block of G*, which coincides with that of G*/Z^G/K.
So it remains only to show that \H \ — ( \ D \ — l)(kB— 1B). By Brauer's Permu-
tation Lemma, H acts transitively on Irr(£))*. In particular for any ι|relrr(0*,
its intertia group Hψ in H is cyclic by Lemma 1 and hence | Irr(T | i/r) | | Irr(Tψ |
ψ) I = |#ψ I by Clifford's Theorem. Using this, we have kB= \ Irr(Γ) | - | Irr(#) |
+ |Irr(Γ|ψ)|=/1 ΓHffψ|. Therefore \H\ = (\Q\-l)\H*\ = (\D\-l)(kB-
1B). This completes the proof of the Theorem.
References
[1] R. Brauer and C. Nesbitt: On the modular characters of groups, Ann. of Math.
42 (1941), 556-590.
[2] P. Fong: Solvable groups and modular representation theory, Trans. Amer. Math.
Soc. 102 (1962), 484-494.
[3] B. Huppert: Zweifach transitive, auflδsbare permutationsgruppen, Math. Z. 68
(1957), 126-150.
[4] G. Michler: On blocks with multiplicity one, preprint.
[5] D.S. Passman: p-Solvable doubly transitive permutation groups, Pacific J. Math.
26 (1968), 555-577.
CARTAN INTEGERS FOR ^-SOLVABLE GROUPS 679
[6] W.F. Reynolds: Protective representations of finite groups in cyclotomic fields,
Illinois J. Math. 9 (1965), 191-198.
[7] Y. Tsushima: On the second reduction theorem of P. Fong, Kumamoto J. Sci.
Math. 13 (1978), 1-5.
Department of Mathematics
Osaka City University
Sumiyoshi-ku, Osaka 558
Japan



A note on Cartan integers for p -solvable groups

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A note on Cartan integers for p -solvable groups

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