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We call a subgroup $\\mathcal{N}_{\\mathfrak{H},\\mathfrak{F}}(G)$ of a finite group $G$ the $\\mathfrak{H}$-$\\mathfrak{F}$-norm of $G$ if $\\mathcal{N}_{\\mathfrak{H},\\mathfrak{F}}(G)$ is the intersection of the normalizers of the products of the $\\mathfrak{F}$-residuals of all subgroups of $G$ and the $\\mathfrak{H}$-radical of $G$. Let $\\pi$ denote a set of primes and let $\\mathfrak{G}_\\pi$ denote the class of all finite $\\pi$-groups. We call the subgroup $\\mathcal{N}_{\\mathfrak{G}_\\pi,\\mathfrak{F}}(G)$ of $G$ the $\\pi\\mathfrak{"},"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":"1308.0948","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2013-08-05T11:44:34Z","cross_cats_sorted":[],"title_canon_sha256":"e918c814ab0d3da8ed2ecd0ad854fa9d13f582803e8e5b2cd1b297366f990f08","abstract_canon_sha256":"2824581b5c650ac36711919798e4ef9b9b51e64fa71a37a8f75dad29dac391d0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:35:57.541229Z","signature_b64":"o/Kj9hQhKtgEKodSB8auNSIj47qMg/O2D8u9K8irq0kc8I3LFi0+8NtaG6E42HBnrA1LkP2QPI4qC/FlUa74DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"786795ea78fd408d1a1022ae7a819a42b86e2cdac84ef4b656610852f0ad913e","last_reissued_at":"2026-05-18T01:35:57.540561Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:35:57.540561Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the $\\pi$$\\mathfrak{F}$-norm and the $\\mathfrak{H}$-$\\mathfrak{F}$-norm of a finite group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Wenbin Guo, Xiaoyu Chen","submitted_at":"2013-08-05T11:44:34Z","abstract_excerpt":"Let $\\mathfrak{H}$ be a Fitting class and $\\mathfrak{F}$ a formation. 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