On generating properties of the weak commutativity of p-groups, p odd

Abstract

In the present paper, we examine the generating properties of Sidki’s weak commutativity group. More precisely, if G and G φ are two isomorphic groups, the weak commutativity group χ(G) is the group generated by G and G φ subject to the relations[g, g φ ] = 1 for all g ∈ G. Here we provide bounds for the number of generators of some subgroups of χ(G) when G is a p-group of odd order and either G is powerful or D(G) is abelian.