Some structural results on the non-abelian tensor square of groups

Abstract

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.

On the q-tensor square of a group

“…It is particularly useful for computer computations of the non-abelian tensor square of polycyclic groups (see e.g. [14,12,1,2,7]). This motivates us to extend the commutator connection of the non-abelian tensor square to study, for any group G, a hat (''power'') and commutator connection of the tensor square modulo q of G, q a non-negative integer, via a group n q ðGÞ.…”

On the size of the third homotopy group of the suspension of an Eilenberg--MacLane space

“…We inform the reader that numerical inequalities of a similar form were recently addressed in [1,14,13] and have motivated us to write the present paper. The purpose of the present paper is in fact a further investigation of the order of the tensor square of nonabelian p -groups and its relations with algebraic topology.…”

The q-tensor square of finitely generated nilpotent groups, q odd

“…In section 2 we extend to G ⊗ q G, q ≥ 0, some structural results found in [5] and [24] concerning G ⊗ G. In section 3 it is established an upper bound for the minimal number of generators of G ⊗ q G when G is a finitely generated nilpotent group of class 2, thus generalizing a result of Bacon found in [2]. We end by computing the q-tensor square of the free nilpotent group of rank n ≥ 2 and class 2, N n,2 , q ≥ 0; this will show, as in the case q = 0 (see [2,Theorem 3.2]), that the cited upper bound is also attained for these groups when q > 1, although in this case N n,2 ⊗ q N n,2 is a non-abelian group.…”