Group actions, deformations, polygroup extensions, and group presentations

TL;DR

This paper extends classical group extension theory to polygroups, introduces a method to derive group presentations from actions, and explores a deformation technique to compute presentations of complex groups like Mathieu groups and Zassenhaus groups.

Contribution

It generalizes extension theory to polygroups, provides a new method for computing group presentations from actions, and develops a deformation approach to analyze and compute presentations of intricate groups.

Findings

Derived explicit presentations for $GL_2$ over valuation rings and fields

Computed presentations for $SL_3$ over arbitrary fields

Presented Zassenhaus groups $M(q^2)$ for odd prime powers q

Abstract

Generalizing classical extension theory, we solve a Schreier-type extension problem for polygroups by groups. As a consequence, we obtain a method for computing a presentation for a group from its action on a set. The usefulness of this method is illustrated by deriving explicit presentations for the groups $GL_2$ over valuation rings and over valued fields, for the groups $SL_3$ over arbitrary fields, as well as for the five Mathieu groups. Moreover, we sketch some aspects of a new deformation technique for groups, their actions, and presentations, and apply it to compute presentations for the sharply $3$-transitive Zassenhaus groups $M(q^2)$ (in the notation of Huppert and Blackburn) for any odd prime power $q$. This computation serves to demonstrate how suitable deformation of groups and their actions interacts with, and thereby enhances, the presentation method.