New Lie products for groups and their automorphisms

TL;DR

This paper introduces new Lie products for groups, generalizing central series, and explores their automorphisms, revealing structural insights in a vast class of finite groups.

Contribution

It develops generalized Lie products for groups, linking derivations to automorphisms, and uncovers new structural properties in large classes of finite groups.

Findings

Uncovered new structures in 4/5 of groups up to size 1000

Linked derivations of Lie rings to group automorphisms

Revealed positive logarithmic proportion of all finite groups exhibit these structures

Abstract

We generalize the common notion of descending and ascending central series. The descending approach determines a naturally graded Lie ring and the ascending version determines a graded module for this ring. We also link derivations of these rings to the automorphisms of a group. This uncovers new structure in 4/5 of the approximately 11.8 million groups of size at most 1000 and beyond that point pertains to at least a positive logarithmic proportion of all finite groups.