{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/universal-covers-of-finite-groups","title":"Universal Covers of Finite Groups","arxiv_id":"1910.11453","date":"2019-10-24","proceeding":null,"authors":["Heiko Dietrich","Alexander Hulpke"],"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.","url_abs":"https://arxiv.org/abs/1910.11453v4","url_pdf":"https://arxiv.org/pdf/1910.11453v4.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"universal-covers-of-finite-groups","repo_url":"https://github.com/hulpke/hybrid","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}