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.