On Fox and augmentation quotients of semidirect products

TL;DR

This paper investigates the structure of augmentation quotients in semidirect products of groups, expressing certain quotient groups via homological and Lie algebra methods, with explicit results for small powers and torsionfree cases.

Contribution

It provides a functorial description of augmentation quotient groups in semidirect products using homological techniques and Lie algebra structures, extending known results to higher powers and torsionfree cases.

Findings

Explicit descriptions for $Q_n(G,H)$ for $n \\le 4$

General formulas for all $n \\ge 2$ under torsionfree conditions

Homological methods relate augmentation quotients to enveloping algebras

Abstract

Let $G$ be a group which is the semidirect product of a normal subgroup $N$ and some subgroup $T$. Let $I^n(G)$, $n\ge 1$, denote the powers of the augmentation ideal $I(G)$ of the group ring $\Z(G)$. Using homological methods the groups $Q_n(G,H) = I^{n-1}(G)I(H)/I^{n}(G)I(H)$, $H=G,N,T$, are functorially expressed in terms of enveloping algebras of certain Lie rings associated with $N$ and $T$, in the following cases: for $n\le 4$ and arbitrary $G,N,T$ (except from one direct summand of $Q_4(G,N)$), and for all $n\ge 2$ if certain filtration quotients of $N$ and $T$ are torsionfree.