The q-tensor square of a powerful p-group

Abstract

Let G be a group and q a non-negative integer. In this work we consider the q-tensor square G⊗qG and the group νq(G), a certain extension of G⊗qG by G×G. Our interest is to study the behavior of these groups under the assumption that G is a powerful finite p-group, p a prime number. Under such assumptions we prove that if exp⁡(G) divides q, then G⊗qG is also powerful and, additionally, d(G⊗qG)≤d(d+1), where d=d(G) denotes the minimal number of generators of G. We also establish bounds for the exponent of G⊗qG in terms of the exponent of G. We derive our results via the embedding of G⊗qG into νq(G). To this end we prove that all terms of the lower central series and of the derived series of νq(G) are powerfully embedded in νq(G), with the only exception of the whole group itself. We give a simple example to show that νq(G) is not necessarily powerful. Our results extend to q≥0 similar results found by Moravec [11] for the non-abelian tensor square G⊗G in the case q=0.