Finiteness conditions for the weak commutativity group and related constructions

The weak commutativity group χ(G) is generated by two isomorphic groups G and Gφ subject to the relations [g,gφ]=1 for all g∈G. We establish a finiteness criterion for the subgroups of χ(G) in terms of the set Tχ(G)={[g1,g2φ]|gi∈G}. We also obtain sufficient conditions under which some specific (local) classes of groups are invariant under the operator χ. Moreover, we prove that if G is a locally finite group with exp (G)=n, then χ(G) is locally finite and has finite n-bounded exponent.