{"paper":{"title":"On the Complexity of Identifying Groups without Abelian Normal Subgroups: Parallel, First Order, and GI-Hardness","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.DS","cs.LO","math.GR"],"primary_cat":"cs.CC","authors_text":"Dan Johnson, Joshua A. Grochow, Michael Levet","submitted_at":"2025-04-28T13:23:46Z","abstract_excerpt":"In this paper, we exhibit an $\\textsf{AC}^{3}$ isomorphism test for groups without Abelian normal subgroups (a.k.a. Fitting-free groups), a class for which isomorphism testing was previously known to be in $\\mathsf{P}$ (Babai, Codenotti, and Qiao; ICALP '12). Here, we leverage the fact that $G/\\text{PKer}(G)$ can be viewed as permutation group of degree $O(\\log |G|)$. As $G$ is given by its multiplication table, we are able to implement the solution for the corresponding instance of Twisted Code Equivalence in $\\textsf{AC}^{3}$.\n  In sharp contrast, we show that when our groups are specified b"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2504.19777","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2504.19777/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}