{"paper":{"title":"On finite groups where the order of every automorphism is a cycle length","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Alexander Bors","submitted_at":"2014-12-29T18:40:43Z","abstract_excerpt":"Using Frobenius normal forms of matrices over finite fields as well as the Burnside Basis Theorem, we give a direct proof of Horo\\v{s}evski\\u{i}'s result that every automorphism $\\alpha$ of a finite nilpotent group has a cycle whose length coincides with $\\mathrm{ord}(\\alpha)$. Also, we give two new sufficient conditions for an automorphism $\\alpha$ of an arbitrary finite group to satisfy this property, namely when $\\mathrm{ord}(\\alpha)$ is a product of at most two prime powers or when $\\alpha$ has a sufficiently large cycle. This will allow us to show that the least order of a group where thi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.8418","kind":"arxiv","version":2},"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"}