Finiteness conditions for the weak commutativity group and related constructions.

The weak commutativity group χ (G) is generated by two isomorphic groups G and G φ subject to the relations [ g , g φ ] = 1 for all g ∈ G . We establish a finiteness criterion for the subgroups of χ (G) in terms of the set T χ (G) = { [ g 1 , g 2 φ ] | g i ∈ 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. Communicated by Mark Lewis [ABSTRACT FROM AUTHOR]