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

Let G be a finite p-group. In this paper we obtain bounds for the exponent of the non-abelian tensor square G circle times G$G \otimes G$ and a certain extension nu(G)$\nu (G)$ of G circle times G$G \otimes G$ by GxG$G \times G$. In particular, we bound exp(nu(G))$\exp (\nu (G))$ in terms of exp(nu(G/N))$\exp (\nu (G/N))$ and exp(N)$\exp (N)$ when G admits some specific normal subgroup N. We also establish bounds for exp(G circle times G)$\exp (G \otimes G)$ in terms of exp(G)$\exp (G)$ and either the nilpotency class or the coclass of the group G, improving some existing bounds.