Date: Fri, 13 Sep 1996 17:28:51 -0600 (CST) From: Giovanni CutoloOn 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