{"paper":{"title":"The classification of normalizing groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"James Mitchell, Jo\\~ao Ara\\'ujo, Max Neunh\\\"offer, Peter J. Cameron","submitted_at":"2012-05-02T15:02:36Z","abstract_excerpt":"Let $X$ be a finite set such that $|X|=n$. Let $\\trans$ and $\\sym$ denote respectively the transformation monoid and the symmetric group on $n$ points. Given $a\\in \\trans\\setminus \\sym$, we say that a group $G\\leq \\sym$ is $a$-normalizing if $$<a,G> \\setminus G=<g^{{-1}}ag\\mid g\\in G>.$$ If $G$ is $a$-normalizing for all $a\\in \\trans\\setminus \\sym$, then we say that $G$ is normalizing. The goal of this paper is to classify normalizing groups and hence answer a question posed elsewhere. The paper ends with a number of problems for experts in groups, semigroups and matrix theory."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1205.0450","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}