{"paper":{"title":"FC-groups with finitely many automorphism orbits","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Alex C. Dantas, Raimundo A. Bastos","submitted_at":"2018-06-28T18:09:06Z","abstract_excerpt":"Let $G$ be a group. The orbits of the natural action of $Aut(G)$ on $G$ are called \"automorphism orbits\" of $G$, and the number of automorphism orbits of $G$ is denoted by $\\omega(G)$. In this paper we prove that if $G$ is an FC-group with finitely many automorphism orbits, then the derived subgroup $G'$ is finite and $G$ admits a decomposition $G = Tor(G) \\times D$, where $Tor(G)$ is the torsion subgroup of $G$ and $D$ is a divisible characteristic subgroup of $Z(G)$. We also show that if $G$ is an infinite FC-group with $\\omega(G) \\leqslant 8$, then either $G$ is soluble or $G \\cong A_5 \\tim"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.11132","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"}