Arch. Math., Vol. 55, 35-37 (1990) 0003-889X/90/5501-0035 $ 2.10/0 �9 1990 Birkhfiuser Verlag, Basel

Two groups with isomorphic group algebras

By

GABRIEL NAVARRO *)

1. Introduction. Let G and H be finite groups. In his paper [4], M. Isaacs proves that if ~G ~ ~H, then the nilpotency of G implies the nilpotency of H, answering a question posed by B. Huppert.

It is well-known that the complex group algebra is a direct irreducible sum of matrix algebras over ~, whose C-dimensions are exactly the degrees of the irreduible complex characters of G. Moreover, ~G, and in general every semisimple algebra, is determined by the knowledge of these dimensions. This is not the case if we replace C by an algebraically closed field with positive characteristic.

Observe that if this characteristic does not divide the order of G, since the group algebra is semisimple and the degrees of its irreducible representations and the degrees of the irreducible C-representations are the same, we will be exactly in Isaacs's hypothesis. In other words, when the characteristic of the field does not divide the order of G, our theorem below is Isaac's Theorem 1 in [4].

Theorem. Let F be an algebraically closed field with char F = p, a prime number. Suppose that FG ~- FH. If for some prime q, G is q-nilpotent, then H is q-nilpotent also.

As by-product of our work, we will obtain the modular version of a theorem of Thompson ([3] 12.2) for p-solvable groups.

It could be remarked that, as we will see, for q a prime distinct of p, the q-nilpotency of G is determined by the knowledge of the degrees of the irreducible Brauer characters and the degrees of the projective indecomposable characters. This.is not the case if q = p (it suffices to consider p = 2, the alternating group on four letters and the cyclic group with 12 elements).

2. The proof. We take notation from [3].

Let p be a prime and IBr (G) be the set of the irreducible p-Brauer characters of G. If ;( E Irr (G), then the restriction to p-regular elements equals Z ~ = ~ dxo ~o. We write

q~ ~ IBr (G)

'~ = ~ dx~o X for the projective indecomposable character afforded by r z e l r r ( G )

Our proof runs parallel to that of Isaacs [4].

*) Research partially supported by DGICYT-proyecto n ~ PS87-0055-C02-02.

3*

36 G NAVARRO ARCH. MATH,

Lemma. Let q =~ p be a prime. Let ~ = {q) e IBr (G) [ q y ~0 (1)}. Then ]G : G' ]q divides Y~ 4o(1 ) go(l), and G has a normal q-complement iff q does not divide

P r o o f . Let N = Oq(G). Let ~f~= {t /elBr(N)[qy~(1) and t/ is G-invariant}. If t/E ~f~, since (o 0/)t/(1 ), q) = 1, using the ideas of ([3], 6.24-6.28), it is straight forward to check that ~/extends to q~ ~ ay.

By VII 9.12 and 9.13 of [21, there are [G: G'iq extensions of I/to G, namely

{2~/such that 2 e Irr (GIG' N)}, and all of them lie in 9 .

Conversely, if r E 9 , by a theorem of Swan ([2] VII 9.20) q)~ ~ ~//~. Moreover, applying a result of W. Willems ([5] 2.8) (#,)N = ~0~.

If we denote by % = { ~ [ q~ e 9 } and % = {4~1 t l e ~/f}, we have that the map ~ ~ (4~o)u from ~r to Vr is well defined.

Since 2 ~e = ~ for 2 ~ IBr (G) linear, we have that if q e ~ and q~s = q, then ~ has [G: G' [q extensions, namely {~o ]2 6 Irr (GIG' N)}, all of them lying in ~/~. Thus

Z ~(1)q~(1) = IG: G'I~ Z ~,0),~(1).

Now, since 4~ = q~,, for g e G, it is clear that

IN[= Z e,(1)~/(1)- 2 e,(l)~l(l)modq. ~/~IBr (N) ~ /~f

Now, G is q-nilpotent iff q does not divide IN] iff q does not divide

Z ~0)e0)/IG: a'l~.

Corollary. Let G be a p-solvable group. Let q 4= p be a prime number. If q divides 9 (1) for all go non linear irreducible p-Brauer character of G, then G has a normal q-complement.

P r o o f. By hypothesis ~ = {(p ~ 1Br(G) [ ~ (1) = 1}.

By Fong's dimensional formula ([1] X.32), we have

Thus q does not divide (I G I~IG: G'I~,)/I G: G'Jq and the previous lemma applies.

N o t e. If H is a p-complement in a finite group G and 2 is a linear character of G, it is rather easy to show that (2.) ~ -- 4~o. Thus, in the above corollary, it suffices to assume that G has a q-complement.

P r o o f o f t h e T h e o r em. First assume that q + p. F G determines the set of its irreducible modules V and of its projective indecomposable modules Pa(V). Since I G: G' I~, is the number of l-dimensional F G-modules to within isomorphism, it follows that I G: G'lq is determined by FG. Now, our lemma applies.

If q = p, since FG determines its Caftan matrix, VII 14.9 of [2], gives us the proof.

Vol. 55, 1990 Two groups with isomorphic group algebras 37

A c k n o w 1 e d g e m e n t s. This work has been done during a stay of the author at Mainz University, supported by a fellowship from Conselleria de Educaci6 i Cultura de la Generalitat Valenciana. He would like to thank also the mathematicians of Mainz for their hospitality. Special thanks to Professor Klaus Doerk.

References

[l] W. FEIT, The Representation Theory of Finite Groups. Amsterdam-New York-Oxford 1982. [2] B. HUPPERT and N. BLACKBURN, Finite Groups II. Berlin-Heidelberg-New York 1982. [3] I. M. ISAACS, Character Theory of Finite Groups. New York-London 1975. [4] I. M. lSAACS, Recovering information about a group from its complex algebra. Arch. Math. 47,

293-295 (1986). [5] W. WILLEMS, On the projectives of a group algebra. Math. Z. 171, 163-174 (1980).

Eingegangen am 22. 11. 1988

Anschrift des Autors:

Gabriel Navarro Departamento de Algebra Facultad de Ciencias Matem~tticas Universitat de Valencia Burjassot, Valencia Spain