The exponent of the non-abelian tensor square and related constructions of $p$-groups

Abstract

Let G be a group. We denote by ν(G) a certain extension of the non-abelian tensor square [G, G ϕ ] by G × G. In this paper we obtain bounds for the exponent of ν(G), when G is a finite p-group. In particular, we prove that if N is a potent normal subgroup of a G, then exp(ν(G)) divides p • exp(N ) • exp(ν(G/N )), where p denotes the prime p if p is odd and 4 if p = 2. As an application, we show that if G is a p-group of maximal class, then exp(ν(G)) divides p 2 • exp(G). We also establish a bound to exp(ν(G)) in terms of exp(G) and the coclass of G. Consequently, we improve some previous bounds to the exponent of Schur Multiplier M (G).