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⊗G$G \otimes G$ and a certain extension ν(G)$\nu (G)$ of G⊗G$G \otimes G$ by G×G$G \times G$. In particular, we bound exp(ν(G))$\exp (\nu (G))$ in terms of exp(ν(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⊗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.