Finiteness conditions for the weak commutativity construction

TL;DR

This paper investigates conditions under which the weak commutativity operator preserves finiteness properties in groups, proving that for locally finite groups with bounded exponent, the operator yields a locally finite group with bounded exponent.

Contribution

It establishes new finiteness conditions for the weak commutativity operator applied to locally finite groups with bounded exponent.

Findings

If G is locally finite with exponent n, then χ(G) is locally finite with exponent dividing n.

The paper provides criteria for the subgroup D(G) to be finite or finitely generated.

It extends known properties of the operator χ to broader classes of groups.

Abstract

The operator, $\chi $, of weak commutativity between isomorphic groups $G$ and $G^{\varphi }$ was introduced by Sidki as \begin{equation*} \chi (G)=\left\langle G \cup G^{\varphi }\mid \lbrack g,g^{\varphi }]=1\,\forall \,g\in G\right\rangle \text{.} \end{equation*} It is known that the operator $\chi $ preserves group properties such as finiteness, solubility and also nilpotency for finitely generated groups. We prove that if $G$ is a locally finite group with $exp(G)=n$, then $\chi(G)$ is locally finite and has finite $n$-bounded exponent. Further, we examine some finiteness criteria for the subgroup $D(G) = \langle [g_1,g_2^{\varphi}] \mid g_i \in G\rangle \leqslant \chi(G)$ in terms of the set $\{[g_1,g_2^{\varphi}] \mid g_i \in G\}$.