We study the non-abelian tensor square G circle times G for the class of groups G that are finitely generated modulo their derived subgroup. In particular, we find conditions on G/G' so that GnG is isomorphic to the direct product of del(G) and the non-abelian exterior square G Lambda G. For any group G, we characterize the non-abelian exterior square G Lambda G in terms of a presentation of G. Finally, we apply our results to some classes of groups, such as the classes of free solvable and free nilpotent groups of finite rank, and some classes of finite p-groups.
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