On the nonabelian tensor squares of free nilpotent groups of finite rank

Abstract

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

The Exterior Squares of Some Crystallographic Groups

“…Kappe et al in [7] investigated on two-generator two-groups of class two and computed the nonabelian tensor squares of these groups. The computations of the nonabelian tensor squares of infinite metacyclic groups and free nilpotent groups of finite rank have been discussed in [8] and [9] respectively. The exterior squares have been computed for finite p  groups of nilpotency class two, infinite nonabelian 2-generator groups of nilpotency class two and symmetric groups of order six in [10], [11] and [12] respectively.…”

Computing the nonabelian tensor squares of polycyclic groups

“…This result has been refined and generalized to all free nilpotent groups of arbitrary class and finite rank in [4] and [3] using the theory presented in this paper.…”

“…with the exception of Lemma 10(ii), can be found in [23] (adjusting for left conjugation) and [4]. We give a proof of Lemma 10(ii) to illustrate how these identities can be derived and omit the proof of Lemma 11, which records additional identities that follow from applications of Lemmas 9 and 10 and routine commutator expansions.…”

On the derived subgroups of the free nilpotent groups of finite rank

“…The following theorem from [1] provides the basic structure for the nonabelian tensor square of a free nilpotent group of finite rank.…”

“…The motivation for this investigation is that the derived subgroup of a free nilpotent group of class c + 1 and rank n is isomorphic to the nonabelian exterior square of the free nilpotent group of class c and rank n. Moreover, the results presented in this paper give complete structure descriptions of the nonabelian tensor squares of free nilpotent groups of finite rank using a result from [1].…”