Date: Fri, 13 Sep 1996 17:28:51 -0600 (CST)
From: Giovanni Cutolo 
On Fri, 13 Sep 1996, David Evans wrote:

> Dear Forum,
> Is it true that if $G$ is any non-trivial finite group then there exists
> a non-split epimorphism $\phi : H \rightarrow G$ for some finite $H$? I
> think this follows from a theorem of Gasch\"utz and some modular
> representation theory, but
> (a) I could be mistaken;
> and
> (b) if I'm not mistaken, could someone provide me with an explicit
> reference for this fact?
>
> David Evans.
>

Dear David,
        a way of proving your claim is to note that for any finite
nontrivial G there exists a finite ZG-module A (Z are the integers) such
that the second dimensional cohomology group H^2(G,A) is non-zero.  Then
you get your group H as any of the non-split extensions of A by G. (In
this way the kernel of the epimorphism is also abelian.)

To show that such A does exists, take a non-trivial cyclic subgroup C of
G, of order n, say. It is easy to construct a finite ZC-module B such that
H^2(C,B) is non-zero. For instance, B could be any finite abelian group
whose order is not coprime with n, with C acting trivially on B (in
this case H^2(C,B)=B/B^n; see e.g., Gruenberg's lecture notes quoted
below, pp. 39-40). Now let A be the coinduced module Hom_{ZC}(ZG,B).
A is finite, and, by Shapiro Lemma (see e.g., Gruenberg's notes again,
p.92), H^2(G,A) is isomorphic to H^2(C,B).

The complete reference to Gruenberg's notes is

Karl Gruenberg: Cohomological Topics in Group Theory, Springer, LNM 143,
1970, (I guess)

Regards,
        Giovanni Cutolo