The relative second Fox and third dimension subgroup of arbitrary groups

TL;DR

This paper provides a comprehensive description of specific relative subgroups within arbitrary groups, focusing on the second Fox and third dimension subgroups using augmentation ideals.

Contribution

It offers a complete characterization of the second relative Fox subgroup and third relative dimension subgroup for any subgroups of an arbitrary group.

Findings

Explicit descriptions of the second relative Fox subgroup.

Explicit descriptions of the third relative dimension subgroup.

General formulas applicable to all subgroups of a group.

Abstract

Let $I_R(G)$ denote the augmentation ideal of the group algebra $R(G)$ of a group $G$ with coefficients in a commutative ring $R$. We give a complete description of the third relative dimension subgroup $G\cap(1+I_R(K)I_R(G)+I^3_R(G))$ and the second relative Fox subgroup $G\cap(1+I_R(K)I_R(H)+I^2_R(G)I_R(H))$ for any subgroups $K$ and $H$ of $G$.