  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 $\{[g_1,g_2^{\varphi}] \mid g_i \in G\}$.
