{"paper":{"title":"On the exponent of a finite group admitting a fixed-point-free four-group of automorphisms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"E. Romano, P. Shumyatsky","submitted_at":"2011-01-27T19:23:54Z","abstract_excerpt":"Let $A$ be a group isomorphic with either $S_4$, the symmetric group on four symbols, or $D_8$, the dihedral group of order 8. Let $V$ be a normal four-subgroup of $A$ and $\\alpha$ an involution in $A\\setminus V$. Suppose that $A$ acts on a finite group $G$ in such a manner that $C_G(V)=1$ and $C_G(\\alpha)$ has exponent $e$. We show that if $A\\cong S_4$ then the exponent of $G$ is $e$-bounded and if $A\\cong D_8$ then the exponent of the derived group $G'$ is $e$-bounded. This work was motivated by recent results on the exponent of a finite group admitting an action by a Frobenius group of auto"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1101.5367","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"}