On Fox quotients of arbitrary group algebras

TL;DR

This paper investigates Fox quotients of group algebras using homological methods, providing explicit computations for low levels and linking them to Lie rings and torsion products.

Contribution

It introduces a homological framework for analyzing Fox quotients of arbitrary group algebras and explicitly computes these quotients for n ≤ 3.

Findings

Explicit formulas for n=2 and n=3 Fox quotients.

Connections established between Fox quotients and enveloping rings of graded Lie rings.

Identification of torsion products of abelian groups in the structure of Fox quotients.

Abstract

For a group $G$, N-series $\cal G$ of $G$ and commutative ring $R$ let $I^n_{R,\cal G}(G)$, $n\ge 0$, denote the filtration of the group algebra $R(G)$ induced by $\cal G$, and $I_R(G)$ its augmentation ideal. For subgroups $H$ of $G$, left ideals $J$ of $R(H)$ and right $H$-submodules $M$ of $I_Z(G)$ the quotients $I_R(G)J/MJ$ are studied by homological methods, notably for $M= I_Z(G)I_Z(H)$, $I_Z(H)I_Z(G) + I_Z([H,G])Z(G)$ and $Z(G)I_Z(N) +I^n_{Z,\cal G}(G)$ with $N \lhd G$ where the group $I_R(G)J/MJ$ is completely determined for $n=2$. The groups $I^{n-1}_{Z,\cal G}(G)I_Z(H)/I^n_{Z,\cal G}(G)I_Z(H)$ are studied and explicitly computed for $n\le 3$ in terms of enveloping rings of certain graded Lie rings and of torsion products of abelian groups.