{"paper":{"title":"On integers that are covering numbers of groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Eric Swartz, Luise-Charlotte Kappe, Martino Garonzi","submitted_at":"2018-05-23T10:49:55Z","abstract_excerpt":"The covering number of a group $G$, denoted by $\\sigma(G)$, is the size of a minimal collection of proper subgroups of $G$ whose union is $G$. We investigate which integers are covering numbers of groups. We determine which integers $129$ or smaller are covering numbers, and we determine precisely or bound the covering number of every primitive monolithic group with a degree of primitivity at most $129$ by introducing effective new computational techniques. Furthermore, we prove that, if $\\mathscr{F}_1$ is the family of finite groups $G$ such that all proper quotients of $G$ are solvable, then"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.09047","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"}