On Non-split Abelian Extensions

Let $$G_{2}$$ be a group which acts trivially on an abelian group $$G_{1}$$ . According to the Schreier’s Theorem, each 2-cocycle $$\varepsilon \in Z^{2}(G_{2},G_{1})$$ determines a group $$G_{\varepsilon }$$ which is a central extension of $$G_{1}$$ by $$G_{2}$$ , and we will denote this group by $$G_{1}\underset{\varepsilon }{\times }G_{2} $$ and call it the perturbed direct product of $$G_{1}$$ by $$G_{2}$$ under $$\varepsilon $$ . The aim of this paper is to study properties of the perturbed direct products. For two distinct 2-cocycles $$\varepsilon _{1}$$ and $$\varepsilon _{2}$$ , we find necessary and sufficient conditions for $$G_{1}\underset{\varepsilon _{1}}{\times }G_{2} $$ to be isomorphic to $$G_{1}\underset{\varepsilon _{2}}{\times }G_{2}$$ . Furthermore, we obtain some results about decompositions for a given perturbed direct product $$G_{1}\underset{ \varepsilon }{\times }G_{2}$$ when $$G_{1}$$ or $$G_{2}$$ is a nontrivial direct product.