The exponent of the non-abelian tensor square and related constructions of $p$-groups
R. Bastos, E. de Melo, N. Gonçalves, C. Monetta
TL;DR
This work targets explicit exponent bounds for the non-abelian tensor square and the related group $\nu(G)$ in finite $p$-groups, linking these bounds to the base group’s exponent, nilpotency class, coclass, and invariants such as $M(G)$, $\mu(G)$, and $\Delta(G)$. The authors develop power–commutator–type control of the lower central series of $\nu(G)$ (notably $\gamma_{i+s+1}(\nu(G)) \leq \gamma_{i+1}(\nu(G))^p$ under suitable conditions) and then translate these structural insights into exponent bounds. They show that $\exp(\nu(G))$ divides $\exp(G)^2 \cdot \exp(M(G))$, with refinements under certain normal-subgroup hypotheses, and obtain sharp bounds for $\exp([G,G^{\varphi}])$ in maximal-class and coclass contexts (e.g., $\exp([G,G^{\varphi}])$ divides $p^2 \cdot \exp(G)$ in maximal-class). Applied to groups of fixed coclass $r$, the results yield that $\exp(M(G))$ and $\exp(\mu(G))$ divide $\exp(G)^{r+1}$ (for odd $p$) or $\exp(G)^{r+3}$ (for $p=2$), improving previous bounds and enhancing computational control over these non-abelian tensor constructions. The findings thus tighten the understanding of how the non-abelian tensor square and related extensions behave across coclass families and maximal-class groups, with implications for group cohomology and homotopy-theoretic interpretations linked to $G$-invariants.
Abstract
Let $G$ be a finite $p$-group. In this paper we obtain bounds for the exponent of the non-abelian tensor square $G \otimes G$ and of $ν(G)$, which is a certain extension of $G \otimes G$ by $G \times G$. In particular, we bound $\exp(ν(G))$ in terms of $\exp(ν(G/N))$ and $\exp(N)$ when $G$ admits some specific normal subgroup $N$. We also establish bounds for $\exp(G \otimes G)$ in terms of $\exp(G)$ and either the nilpotency class or the coclass of the group $G$, improving some existing bounds.