Restricted Lie algebras of polycyclic groups, II

TL;DR

This paper investigates the structure of restricted Lie algebras associated with polycyclic groups, providing conditions for their abelian or free abelian nature and developing methods for constructing relevant series.

Contribution

It introduces new conditions for the existence of specific p-series in polycyclic groups and develops methods based on filtrations and valuations for their construction.

Findings

Conditions for p-series with abelian restricted Lie algebras

Methods for constructing filtrations and valuation functions

Applications to the structure of polycyclic groups

Abstract

This is the second paper in the series of three. We study restricted Lie algebras of polycyclic groups and obtain conditions for existence of $p$-series with associated restricted Lie algebra abelian or free abelian with rank equal to the Hirsch number of the group. We develop methods for constructing such series, these methods are based on construction of filtrations and valuation functions in the group rings.