Finiteness conditions for the weak commutativity construction

Abstract

The operator, $\chi$, of weak commutativity between isomorphic groups $G$ and $G^{\phi }$ was introduced by Sidki as \begin{equation*} χ(G)=\left\langle G \cup G^{φ}\mid \lbrack g,g^{φ}]=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 $D(G)=⟨[g_1,g_2^{\phi }]\mid g_i\in G⟩⩽\chi (G)$ in terms of the set $\{[g_1,g_2^{\phi }]\mid g_i\in G\}$.