On the non-abelian tensor square of groups of order dividing $p^{5}$

TL;DR

This paper provides explicit structures of various algebraic constructs like the non-abelian tensor square, exterior square, and related centers for all groups of order dividing p^5, advancing understanding of their algebraic topology and group theory.

Contribution

It offers a comprehensive and explicit characterization of the non-abelian tensor and exterior squares for all groups of order dividing p^5, which was previously not fully detailed.

Findings

Explicit structures of non-abelian tensor squares for these groups.

Descriptions of the third homotopy group of certain spaces associated with these groups.

Characterization of tensor and exterior centers for groups of order dividing p^5.

Abstract

In this paper we consider all groups of order dividing $p^5$. We obtain the explicit structure of the non-abelian tensor square, non-abelian exterior square, tensor center, exterior center, the third homotopy group of suspension of an Eilenberg-MacLain space $k(G,1) $ and $\triangledown(G)$ of such groups.