A note on semidirect products and nonabelian tensor products of groups

Abstract

Let \(G\) and \(H\) be groups which act compatibly on one another. In [2] and [8] it is considered a group construction \(\eta(G,H)\) which is related to the nonabelian tensor product \(G \otimes H\). In this note we study embedding questions of certain semidirect products \(A \rtimes H\) into \(\eta(A, H)\), for finite abelian \(H\)-groups \(A\). As a consequence of our results we obtain that complete Frobenius groups and affine  groups over finite fields are embedded into \(\eta(A, H)\) for convenient groups \(A\) and \(H\). Further, on considering finite metabelian groups \(G\) in which the derived subgroup has order coprime with its index we establish the order of the nonabelian tensor square of \(G\).