The nonabelian product modulo sum

TL;DR

This paper demonstrates that for nonabelian groups without involutions, the topologist's product modulo finite words is uniquely determined up to isomorphism, unlike the abelian case where it depends on the sequence.

Contribution

It establishes a fundamental difference between abelian and nonabelian groups regarding the structure of their topologist's products modulo sum.

Findings

Nonabelian groups without involutions have sequence-independent topologist's products.

Contrasts with abelian groups where the product's isomorphism class depends on the sequence.

Shows structural rigidity in nonabelian case versus flexibility in abelian case.

Abstract

It is shown that if $\{H_n\}_{n \in \omega}$ is a sequence of groups without involutions, with $1 < |H_n| \leq 2^{\aleph_0}$, then the topologist's product modulo the finite words is (up to isomorphism) independent of the choice of sequence. This contrasts with the abelian setting: if $\{A_n\}_{n \in \omega}$ is a sequence of countably infinite torsion-free abelian groups, then the isomorphism class of the product modulo sum $\prod_{n \in \omega} A_n/\bigoplus_{n \in \omega} A_n$ is dependent on the sequence.