Universal Covers of Finite Groups

TL;DR

This paper introduces a method to compute group extensions using covering groups, Fox derivatives, and a new hybrid format, enabling detailed analysis of finite groups and their quotients with applications in computational group theory.

Contribution

It provides a novel approach to compute extensions of finite groups via covering groups, incorporating Fox derivatives and a hybrid presentation format for computational purposes.

Findings

Describes how to compute extensions using covering groups and modules.

Proves Fox derivatives can be represented via wreath products.

Introduces a hybrid format for computer representation of group extensions.

Abstract

Motivated by quotient algorithms, such as the well-known $p$-quotient or solvable quotient algorithms, we describe how to compute extensions $\tilde H$ of a finite group $H$ by a direct sum of isomorphic simple $\mathbb{Z}_p H$-modules such that $H$ and $\tilde H$ have the same number of generators. Similar to other quotient algorithms, our description will be via a suitable covering group of $H$. Defining this covering group requires a study of the representation module, as introduced by Gasch\"utz in 1954. Our investigation involves so-called Fox derivatives (coming from free differential calculus) and, as a by-product, we prove that these can be naturally described via a wreath product construction. An important application of our results is that they can be used to compute, for a given epimorphism $G\to H$ and simple $\mathbb{Z}_p H$-module $V$, the largest quotient of $G$ that maps onto $H$ with kernel isomorphic to a direct sum of copies of $V$. For this we also provide a description of how to compute second cohomology groups for the (not necessarily solvable) group $H$, assuming a confluent rewriting system for $H$. To represent the corresponding group extensions on the computer, we introduce a new hybrid format that combines this rewriting system with the polycyclic presentation of the module.