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Furthermore, considering, for exponents $e\\in\\left(0,1\\right)$, the condition that a finite group $G$ have an automorphism with a cycle of length at least $|G|^e$, such a condition is strong enough to imply $|\\operatorname{Rad}(G)|\\to\\in"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1501.07172","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-01-28T16:16:21Z","cross_cats_sorted":[],"title_canon_sha256":"79b34b5d40b9f141c0bf78cb0b2383b47c96ec0daf3464abbcbca8b7db9e59e1","abstract_canon_sha256":"dbb05f2cf32954cf10a8e89d021c7c7ce52ab49e76206880ad4f4d09fe6e88e5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:51:25.006037Z","signature_b64":"vw9djeR4EJ4TkuqoOBiBOB7BPJn0FQ8wk8eWMd+4n8oOL/eoNT+BX0padochp0bMR2fjhpKdLR+aWDIyL4haDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ccf235d631918f652c23aab56f1cd0ecf96c9b546b147ddc640245ffa0a0d361","last_reissued_at":"2026-05-18T01:51:25.005511Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:51:25.005511Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Cycle lengths in finite groups and the size of the solvable radical","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Alexander Bors","submitted_at":"2015-01-28T16:16:21Z","abstract_excerpt":"We prove the following: For any $\\rho\\in\\left(0,1\\right)$, if a finite group $G$ has an automorphism with a cycle of length at least $\\rho\\cdot|G|$, then the index of the solvable radical $\\operatorname{Rad}(G)$ in $G$ is bounded from above in terms of $\\rho$, and such a condition is strong enough to imply solvability of $G$ if and only if $\\rho>\\frac{1}{10}$. 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