On Non-split Abelian Extensions

Let G2 be a group which acts trivially on an abelian group G1 . According to the Schreier’s Theorem, each 2-cocycle ε∈Z2(G2,G1) determines a group Gε which is a central extension of G1 by G2 , and we will denote this group by G1×εG2 and call it the perturbed direct product of G1 by G2 under ε . The aim of this paper is to study properties of the perturbed direct products. For two distinct 2-cocycles ε1 and ε2 , we find necessary and sufficient conditions for G1×ε1G2 to be isomorphic to G1×ε2G2 . Furthermore, we obtain some results about decompositions for a given perturbed direct product G1×εG2 when G1 or G2 is a nontrivial direct product.