Non abelian tensor square of non abelian prime power groups

TL;DR

This paper improves bounds on the order of the tensor square of non-abelian p-groups and describes their structure when bounds are tight, also relating to the homotopy group order.

Contribution

It refines existing bounds for non-abelian p-groups' tensor squares and characterizes the groups achieving these bounds, extending to topological invariants.

Findings

Improved upper bounds for tensor square order of non-abelian p-groups.

Structural description of p-groups when bounds are attained.

New bounds for the order of _3(SK(G,1)).

Abstract

For every $p$-group of order $p^n$ with the derived subgroup of order $p^m$, Rocco in \cite{roc} has shown that the order of tensor square of $G$ is at most $p^{n(n-m)}$. In the present paper not only we improve his bound for non-abelian $p$-groups but also we describe the structure of all non-abelian $p$-groups when the bound is attained for a special case. Moreover, our results give as well an upper bound for the order of $\pi_3(SK(G, 1))$.