On the non-albelian tensor square of a nilpotent group of class two

Abstract

The nonabelian tensor square G⊗G of a group G is generated by the symbols g⊗h, g, h ∈ G, subject to the relations,for all g, g′, h, h′ ∈ G, where The tensor square is a special case of the nonabelian tensor product which has its origins in homotopy theory. It was introduced by R. Brown and J. L. Loday in [4] and [5], extending ideas of Whitehead in [6].

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.…”

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

“…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…”

Certain homological functors of 2-generator 𝑝-groups of class 2

“…This last step can be difficult, and can sometimes lead to the incorrect identification of the nonabelian tensor square with a group that is in fact a proper quotient of it (e.g., see the comments after the proof of Theorem 4.4 in [21]). The crossed pairing method was used in [2] to determine the nonabelian tensor square of the free nilpotent groups of class two and finite rank, and in [5,9] for the free 2-Engel groups of finite rank; the latter is particularly computationally intensive. They were also behind the attempts for the 2-generator p-groups of class two [3,4,21].…”