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 η(G,H) which is related to the nonabelian tensor product G⊗H. In this note we study embedding questions of certain semidirect products A⋊H into η(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 η(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.