Abstract. We study the non-abelian tensor square G n G for the class of groups G that are finitely generated modulo their derived subgroup. In particular, we find conditions on G=G 0 so that G n G is isomorphic to the direct product of 'ðGÞ and the non-abelian exterior square G^G. For any group G, we characterize the non-abelian exterior square G^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.

In this paper we develop a theory for computing the nonabelian tensor square and related computations for finitely presented groups and specialize it to polycyclic groups. This theory provides a framework for making nonabelian tensor square computations for polycyclic groups and is the basis of an algorithm for computing the nonabelian tensor square for any polycyclic group.

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We determine the nonabelian tensor squares and related homological functors of the free nilpotent groups of finite rank.

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