On some series of a group related to the non-abelian tensor square of groups

TL;DR

This work analyzes the group $\nu(G)$, an extension of the non-abelian tensor square $G \otimes G$ by $G \times G$, to obtain a detailed description of its derived and lower central series. Central to the approach are the three normal subgroups $\Upsilon_1(G)=[G,G^{\varphi}]$, $\Upsilon_2(G)=[\Theta(G),G]$, and $\Upsilon_3(G)=[\Theta(G),G^{\varphi}]$, which are all isomorphic to $G \otimes G$ and form a central product decomposition of $\nu(G)'$. The paper proves precise formulas for the derived and lower central series in terms of these subgroups, and demonstrates how $\nu(G)'$ can be computed in several important cases where $G \otimes G$ has a known decomposition, yielding explicit structures such as $\nu(G)'\cong\mu(G)\times G'\times G'\times G'$ under suitable hypotheses. It also derives exponent bounds for $\nu(G)$ and its sections, connects to the non-abelian exterior square via $\tau(G)=\nu(G)/\Delta(G)$, and provides concrete examples including free groups, linear groups over finite fields, and metacyclic groups. Overall, the work offers a unified framework to compute $\nu(G)'$, analyze the growth of its lower central series, and bound exponents in broad classes of groups, with potential implications for computational group theory and homological invariants.

Abstract

Let $G$ be a group. We denote by $ν(G)$ a certain extension of the non-abelian tensor square $G \otimes G$ by $G \times G$. In this paper we prove that the derived subgroup $ν(G)'$ is a central product of three normal subgroups of $ν(G)$, all isomorphic to the non-abelian tensor square $G \otimes G$. As a consequence, we describe the structure of each term of the derived and lower central series of the group $ν(G)$.

Theorems & Definitions (16)

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