On the q-tensor square of a group

Abstract

We study the non-abelian tensor square modulo q of a group, where q is a nonnegative integer, via an operator n q in the class of groups. Structural properties and finiteness conditions of n q ðGÞ are investigated. We compute the non-abelian tensor square modulo q of cyclic groups and develop a theory for computing n q ðGÞ and some of its relevant sections for polycyclic groups G. This extends the existing theory from the case q ¼ 0 to all nonnegative integers q. Additionally, a table of examples is produced with the help of the GAP system.

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

“…By [10, Proposition 2.9] Υ q (G) is isomorphic to the q-tensor square G ⊗ q G, for all q ≥ 0. We then get a result (see [10,Corollary 2.11]) analogous to one due to Ellis in [14]: ν q (G) ∼ = G (G (G ⊗ q G)); this generalizes a similar result found in [23] for q = 0.…”

“…For a finitely generated abelian group A, its q-tensor square Υ q (A) can be computed by repeated applications of the following two results from [10].…”

“…Following [10] we write θ q (G) = Ker(ρ) and µ q (G) = Ker(ρ ) = Υ q (G) ∩ θ q (G). It follows that Υ q (G)/µ q (G) ∼ = G G q .…”

“…It is a well known fact (see [23]) that the subgroup Υ(G) = [G, G ϕ ] of ν(G) is isomorphic to the non-abelian tensor square G ⊗ G, as defined by Brown and Loday in their seminal paper [8]. A modular version of the operator ν was considered in [10], where for any non-negative integer q the authors introduced and studied a group ν q (G), which in turn is an extension of the so called q-tensor square of G, G⊗ q G, first defined by Conduché and Rodriguez-Fernandez in [11] (see also [14], [7]). In order to describe the group ν q (G), if q ≥ 1 then let G = { k | k ∈ G} be a set of symbols, one for each element of G (for q = 0 we set G = ∅, the empty set).…”

“…The extension of the existing theory from q = 0 to all non-negative integers q, as addressed for instance in [10], broadens the scope of these connections, now in a hat ("power") and commutator approach to the q-tensor square, q ≥ 0.…”