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It is proved that if $G$ admits a soluble group of automorphisms $A$ of coprime order, then $h^*(G)$ is bounded in terms of $h^* (C_G(A))$, where $C_G(A)$ is the fixed-point subgroup, and the number of prime factors of $|A|$ counting multiplicities. The result follows from the special case when $A=\\langle\\varphi\\rangle$ is of prime order, where it is p"},"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":"1405.1946","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2014-05-03T13:05:39Z","cross_cats_sorted":[],"title_canon_sha256":"fb302a842177aee62ba1805d3792d31b40ee35a1ac063443deda07a21b3afd37","abstract_canon_sha256":"91eb6eda65ec44643518b5628743aaff3f2fe7f8099fdfa794af011fce3c5137"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:28:23.014835Z","signature_b64":"5DK76RLawCVmbymYpzZtI9uwuOD/lXBvAJMihhrEVSyRYOFcqPNhDpl2hv9go57qkoiVUOiwnqQJEcJtrwmYDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"55e522c449618a1f52df09b35195766d1cde3d50a787586b12d1a9b757c6e577","last_reissued_at":"2026-05-18T02:28:23.014278Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:28:23.014278Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the length of finite groups and of fixed points","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"E. I. Khukhro, P. Shumyatsky","submitted_at":"2014-05-03T13:05:39Z","abstract_excerpt":"The generalized Fitting height of a finite group $G$ is the least number $h=h^*(G)$ such that $F^*_h(G)=G$, where the $F^*_i(G)$ is the generalized Fitting series: $F^*_1(G)=F^*(G)$ and $F^*_{i+1}(G)$ is the inverse image of $F^*(G/F^*_{i}(G))$. It is proved that if $G$ admits a soluble group of automorphisms $A$ of coprime order, then $h^*(G)$ is bounded in terms of $h^* (C_G(A))$, where $C_G(A)$ is the fixed-point subgroup, and the number of prime factors of $|A|$ counting multiplicities. 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