On Computing the Non-Abelian Tensor Squares of the Free 2-Engel Groups

Abstract

In this paper we compute the non-abelian tensor square for the free 2-Engel group of rank n > 3. The non-abelian tensor square for this group is a direct product of a free abelian group and a nilpotent group of class 2 whose derived subgroup has exponent 3. We also compute the non-abelian tensor square for one of the group's finite homomorphic images, namely, the Burnside group of rank n and exponent 3.

The nonabelian tensor square of Bieberbach group of dimension five with dihedral point group of order eight

“…We can see that the nonabelian tensor square of the first Bieberbach group of dimension five with dihedral point group of order eight, B 1 (5), is generated by ten elements. This results can further be used to find other useful properties of B 1 (5) such as the homological functors of the group.…”

“…There are some methods that have been used by earlier researchers in computing the nonabelian tensor square G G of a group G. In [3][4][5] they used crossed pairing method to compute the nonabelian tensor square by determining a unique homomorphism *: G G L . Brown et al in [6] used the definition given in (1) to…”

“…Beside the commutator calculus, the following lemmas about the structural results of the group ) (G are used in the computation of the nonabelian tensor square of 1 (5). B Lemma 1 [9,10].…”

“…The main focus of this paper is to compute the nonabelian tensor square of the first Bieberbach group of dimension five with dihedral point group of order eight, denoted by 1 (5). B The result has been verified using Group, Algorithms and Programming (GAP) software.…”

Computing the nonabelian tensor squares of polycyclic groups

“…The original motivation for this paper came from attempting to compute the nonabelian tensor squares of the free nilpotent groups of class 3 and finite rank. The crossed pairing method used to compute the nonabelian tensor squares of the free 2-Engel groups of finite rank is overwhelmingly complex (see [5]) and the prospect of even more complexity for the class 3 case made the project untenable. However, using the methods developed in this paper, we obtain the following result with a modest amount of effort.…”