Homotopy theory and generalized dimension subgroups

TL;DR

This paper explores a homotopy-theoretic approach to generalized dimension subgroups in group theory, revealing their structure and properties through topological methods.

Contribution

It introduces a homotopy-theoretic framework for analyzing generalized dimension subgroups and identifies their quotient with a subgroup of a kernel in homotopy theory.

Findings

The generalized dimension subgroup has exponent 2 modulo the commutator subgroup.

The quotient is identified with a subgroup of the kernel of the Hurewicz homomorphism.

Homotopy theory provides a natural setting for understanding these algebraic structures.

Abstract

Let $G$ be a group and $R,S,T$ its normal subgroups. There is a natural extension of the concept of commutator subgroup for the case of three subgroups $\|R,S,T\|$ as well as the natural extension of the symmetric product $\|\bf r,\bf s,\bf t\|$ for corresponding ideals $\bf r,\bf s, \bf t$ in the integral group ring $\mathbb Z[G]$. In this paper, it is shown that the generalized dimension subgroup $G\cap (1+\|\bf r,\bf s,\bf t\|)$ has exponent 2 modulo $\|R,S,T\|.$ The proof essentially uses homotopy theory. The considered generalized dimension quotient of exponent 2 is identified with a subgroup of the kernel of the Hurewicz homomorphism for the loop space over a homotopy colimit of classifying spaces.