On Fox and augmentation quotients of semidirect products

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 $\mathbb{Z}(G)$. 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.

arXiv comment39 pages; paper thoroughly revised: notation and presentation improved, many details and new result added (Theorem 1.7)

References in corpus (1)

• The relative second Fox and third dimension subgroup of arbitrary groups