{"paper":{"title":"Exponent of a finite group admitting a coprime automorphism","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Pavel Shumyatsky, Sara Rodrigues","submitted_at":"2019-07-04T13:34:00Z","abstract_excerpt":"Let $G$ be a finite group admitting a coprime automorphism $\\phi$ of order $n$. Denote by $G_{\\phi}$ the centralizer of $\\phi$ in $G$ and by $G_{-\\phi}$ the set $\\{ x^{-1}x^{\\phi}; \\ x\\in G\\}$. We prove the following results.\n  1. If every element from $G_{\\phi}\\cup G_{-\\phi}$ is contained in a $\\phi$-invariant subgroup of exponent dividing $e$, then the exponent of $G$ is $(e,n)$-bounded.\n  2. Suppose that $G_{\\phi}$ is nilpotent of class $c$. If $x^{e}=1$ for each $x \\in G_{-\\phi}$ and any two elements of $G_{-\\phi}$ are contained in a $\\phi$-invariant soluble subgroup of derived length $d$,"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.02396","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"}