{"paper":{"title":"Covers and Normal Covers of Finite Groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Andrea Lucchini, Martino Garonzi","submitted_at":"2013-10-07T13:23:15Z","abstract_excerpt":"For a finite non cyclic group $G$, let $\\gamma(G)$ be the smallest integer $k$ such that $G$ contains $k$ proper subgroups $H_1,\\dots,H_k$ with the property that every element of $G$ is contained in $H_i^g$ for some $i \\in \\{1,\\dots,k\\}$ and $g \\in G.$ We prove that if $G$ is a noncyclic permutation group of degree $n,$ then $\\gamma(G)\\leq (n+2)/2.$ We then investigate the structure of the groups $G$ with $\\gamma(G)=\\sigma(G)$ (where $\\sigma(G)$ is the size of a minimal cover of $G$) and of those with $\\gamma(G)=2.$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.1775","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":""},"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"}