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

TL;DR

This paper establishes new bounds for the exponent of the non-abelian tensor square and related constructions of finite p-groups, linking these bounds to group invariants like nilpotency class and coclass.

Contribution

It provides improved bounds for the exponent of the non-abelian tensor square and related groups, extending previous results and relating these bounds to group structure and normal subgroups.

Findings

Bounds for $ ext{exp}( u(G))$ in terms of $ ext{exp}( u(G/N))$ and $ ext{exp}(N)$

Improved bounds for $ ext{exp}(G ensor G)$ based on $ ext{exp}(G)$, nilpotency class, and coclass

Enhanced understanding of the relationship between group invariants and exponents of tensor constructions

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 $\nu(G)$, which is a certain extension of $G \otimes G$ by $G \times G$. In particular, we bound $\exp(\nu(G))$ in terms of $\exp(\nu(G/N))$ 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.