Compatible filters with isomorphism testing

TL;DR

This paper investigates conditions under which filters of groups produce a meaningful graded Lie ring that accurately reflects the group's structure, enabling isomorphisms to be transferred between groups and Lie rings.

Contribution

It identifies specific properties of filters that ensure a bijection between the group and its associated graded Lie ring, establishing when isomorphisms correspond.

Findings

Conditions for filters where the Lie ring and group are in bijection

Proof that isomorphisms between groups are induced by Lie ring isomorphisms under these conditions

Analysis of cases where the associated Lie ring is trivial or arbitrarily large

Abstract

Like the lower central series of a nilpotent group, filters generalize the connection between nilpotent groups and graded Lie rings. However, unlike the case with the lower central series, the associated graded Lie ring may share few features with the original group: e.g. the associated Lie ring may be trivial or arbitrarily large. We determine properties of filters such that the Lie ring and group are in bijection. We prove that, under such conditions, every isomorphism between groups is induced by an isomorphism between graded Lie rings.