On the isomorphism problem for split extensions

Abstract

In this paper, we study an important well-known structure operation, namely, the semidirect product (or split extensions) which is a very useful tool to structure certain kinds of groups. More precisely, we study the isomorphism problem for semidirect products and then we determine how isomorphism of semidirect products and conjugacy of the images of the corresponding actions are related. As an application, for two positive integers [Formula: see text] and [Formula: see text], we compute the number of upper isomorphism classes of split extensions of an elementary abelian [Formula: see text]-group of order [Formula: see text] by an elementary abelian [Formula: see text]-group of order [Formula: see text]. Furthermore, we deal with split extensions where the kernel is a non-abelian [Formula: see text]-group of nilpotency class two and the quotient is an elementary abelian [Formula: see text]-group.

On the Isomorphism Problem for Central Extensions II

“…More precisely, we complete the work with the isomorphism problem for central extensions in different special cases. So most of the results obtained here considered as a generalization of those in [9,10].…”

“…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.…”

“…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. As is well known, the identity element of the second cohomology group corresponds to the split extensions.…”

“…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.…”