On non-split abelian extensions II

Abstract

Let [Formula: see text] be a group acting on an abelian group [Formula: see text] via a homomorphism [Formula: see text] and let a 2-cocycle [Formula: see text]. By Schreier’s theorem, the pair [Formula: see text] determines a group [Formula: see text] which can arise as a non-split extension of [Formula: see text] by [Formula: see text], denoted by [Formula: see text] and called the perturbed semidirect product of [Formula: see text] by [Formula: see text] under [Formula: see text]. In this paper, we classify the perturbed semidirect products and give some of their properties. Furthermore, we find necessary and sufficient conditions for two perturbed semidirect products to be isomorphic.

On the Isomorphism Problem for Central Extensions II

“…As mentioned in the introduction, our choice to focus on the isomorphism problem for extensions with abelian kernel group is partially motivated by the Jordan-Hölder Theorem. Our other works of particular relevance are [8][9][10][11]. We study in [11] the isomorphism problem for split extensions, and as an application, we determine how isomorphism of split extensions and conjugacy of the images of the corresponding actions are related.…”

“…This motivates us to consider also the non-identity different elements, which give rise to the construction of non-split extensions. So, in [8][9][10], we study the isomorphism problem for non-split abelian extensions in some special cases. We mainly deal with isomorphisms leaving one of the two factors or even both invariant.…”

“…However, this answer will not enable us to construct all possible non-isomorphic central extensions of G 1 by G 2 (the isomorphism problem). In fact, it is very hard to solve the isomorphism problem, but it has been discussed for some special cases in [8][9][10]. In fact, those results do not concern general isomorphisms, but only those of certain type, namely leaving the kernel group or both the two factors invariant, inducing the identity or a commuting automorphism on the quotient group.…”

On the isomorphism problem for split extensions

“…Very recently, we study in [12] the isomorphism problem for split extensions. In [11,13], we characterize the isomorphism problem for non-split abelian extensions. In fact, most of the results of those studies do not concern general isomorphisms, but only those of certain type, namely leaving one of the two factors or even both invariant.…”

“…However, this answer will not enable us to construct all possible non-isomorphic central extensions of G 1 by G 2 (the isomorphism problem). In fact, it is very hard to solve the isomorphism problem, but it has been discussed for some special cases in [8][9][10]. In fact, those results do not concern general isomorphisms, but only those of certain type, namely leaving the kernel group or both the two factors invariant, inducing the identity or a commuting automorphism on the quotient group.…”