Universal covers of finite groups

Heiko Dietrich, Alexander Hulpke

Research output: Contribution to journalArticleResearchpeer-review

Abstract

Motivated by the success of quotient algorithms, such as the well-known p-quotient or solvable quotient algorithms, in computing information about finite groups, we describe how to compute finite extensions H˜ of a finite group H by a direct sum of isomorphic simple ZpH-modules such that H and 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 Gaschütz 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→H and simple ZpH-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 a computer, we introduce a new hybrid format that combines this rewriting system with the polycyclic presentation of the module.

Original languageEnglish
Pages (from-to)681-712
Number of pages32
JournalJournal of Algebra
Volume569
DOIs
Publication statusPublished - 1 Mar 2021

Keywords

  • Cohomology of groups
  • Computational group theory
  • Quotient algorithms

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