From: Bettina EickDate: Tue, 22 Apr 1997 17:41:41 +0200 (MET DST) >Dear Forum, > >It is well known that the second cohomology group H^2(G,A) >classifies extensions of G by A. >When two elements of H^2(G,A) >corresponds to nonisomorphic groups? > >Alexei Davydov I don't have a general solution for this question. There is the following partial solution to the problem. Suppose A is elementary abelian. Let H_1 and H_2 be two extensions of G by A. Let A_i < H_i be the subgroup corresponding to A in H_i for i = 1,2. Then we call H_1 and H_2 stongly isomorphic, iff there exists an isomorphism H_1 -> H_2 which maps A_1 onto A_2. The stong isomorphism classes of extensions can be classified. Let phi : G -> Aut(A) the operation homomorphism. We call a tuple ( a, b ) of Aut(G) x Aut( M ) a compatible pair, iff phi(g^a) = phi(g)^b for all g in G. The subgroup of all compatible pairs is denoted by C. There exists an action of C on H^2(G,A). Let psi in Z^2(G,A) and (a,b) in C. Then we define psi^(a,b)(g,h) := psi(g^(a^-1), h^(a^-1))^b. This defines an action of C on Z^2(G,A) which learves B^2(G,A) invariant. Thus we obtain an action of C on H^2(G, A). (This action is linear.) The strong isomorphism classes of extensions correspond one-to-one to the orbits of C on H^2(G, A) under this action. Best wishes, Bettina Eick