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Let $f(x) = a_0 + a_1 \\cdot x + \\cdots + a_d \\cdot x^d \\in \\mathbb{Z}[x]$ be a primitive polynomial and let $G$ be a finite group with an automorphism $\\varphi$ of coprime order satisfying $ g^{a_0} \\cdot \\varphi(g)^{a_1} \\cdots \\varphi^d(g)^{a_d} = 1 $, for all $g \\in G$. Then the soluble radical of $G$ has $(d,|C_G(\\varphi)|)$-boundex Fitting height and index. The bounds are made expli"},"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":"2202.10087","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2022-02-21T09:58:45Z","cross_cats_sorted":[],"title_canon_sha256":"13e3effa1c7d92582bffbe1ec3c56e3ab452817c758434cd882cf7ae78a2bfc1","abstract_canon_sha256":"fe211a7363ee1d105a099f037e9f8a99b965839ab54f5d8acc27b7de91d943be"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:58:38.693311Z","signature_b64":"dFu1TakQS5pshOLL+NslLXl4H+IZ1Qmvj7pAhfiCCmRbSGD6NzCzveG/J0Q9cO7BEbm+vRnPbHcA3elQXkJNCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1036dbb1ce997288de9f0734500ac1b686151bfafbcb96863245aafe34367573","last_reissued_at":"2026-07-05T03:58:38.692958Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:58:38.692958Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Finite groups admitting a coprime automorphism satisfying an additional polynomial identity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Wolfgang Alexander Moens","submitted_at":"2022-02-21T09:58:45Z","abstract_excerpt":"It is known that a finite group with an automorphism $\\varphi$ of coprime order has a soluble radical of $(|\\varphi|,|C_G(\\varphi)|)$-bounded Fitting height and index. We extend this classic result as follows. Let $f(x) = a_0 + a_1 \\cdot x + \\cdots + a_d \\cdot x^d \\in \\mathbb{Z}[x]$ be a primitive polynomial and let $G$ be a finite group with an automorphism $\\varphi$ of coprime order satisfying $ g^{a_0} \\cdot \\varphi(g)^{a_1} \\cdots \\varphi^d(g)^{a_d} = 1 $, for all $g \\in G$. Then the soluble radical of $G$ has $(d,|C_G(\\varphi)|)$-boundex Fitting height and index. 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