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We say that $H$ satisfies partial $\\Pi$-property in $G$ if there exists a chief series $\\mathit{\\Gamma}_G:1=G_0<G_1<\\cdots<G_n=G$ of $G$ such that for every $G$-chief factor $G_i/G_{i-1}$ ($1\\leq i\\leq n$) of $\\mathit{\\Gamma}_G$, $|G/G_{i-1}:N_{G/G_{i-1}}(HG_{i-1}/G_{i-1}\\cap G_i/G_{i-1})|$ is a $\\pi(HG_{i-1}/G_{i-1}\\cap G_i/G_{i-1})$-number. Our main results are listed here:\n  Theorem A. Let $\\mathfrak{F}$ be a solubly saturated formation containing $\\mathfrak{U}$ and $E$ a normal subgroup of $G$ with $G/E\\in \\mathfrak{F}$. Let $X\\unlhd G$ such tha"},"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":"1301.6361","kind":"arxiv","version":9},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2013-01-27T14:46:12Z","cross_cats_sorted":[],"title_canon_sha256":"a42d6c9b0ccc3dc98b99ed6949e676613b2423c6eb3b4edc602256fda113b1b7","abstract_canon_sha256":"e2ec2ded42310e999a73f370293caa346890290784fe5d1ceae1f0557785000d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:38:44.653650Z","signature_b64":"3FRQCf2Iknho0YDTEgjkIaTRRJSjBZzazWus8ytCodQ5JIhCXb6wIS1dedDksrAZnkfz7DiQKzC02M+NE72iBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"515dbb5e3a32d1be1102782de27c5be3f7128200ec761a2e5b425ce143f59c1b","last_reissued_at":"2026-05-18T02:38:44.653269Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:38:44.653269Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On partial $\\Pi$-property of subgroups of finite groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Wenbin Guo, Xiaoyu Chen","submitted_at":"2013-01-27T14:46:12Z","abstract_excerpt":"Let $H$ be a subgroup of a finite group $G$. We say that $H$ satisfies partial $\\Pi$-property in $G$ if there exists a chief series $\\mathit{\\Gamma}_G:1=G_0<G_1<\\cdots<G_n=G$ of $G$ such that for every $G$-chief factor $G_i/G_{i-1}$ ($1\\leq i\\leq n$) of $\\mathit{\\Gamma}_G$, $|G/G_{i-1}:N_{G/G_{i-1}}(HG_{i-1}/G_{i-1}\\cap G_i/G_{i-1})|$ is a $\\pi(HG_{i-1}/G_{i-1}\\cap G_i/G_{i-1})$-number. Our main results are listed here:\n  Theorem A. Let $\\mathfrak{F}$ be a solubly saturated formation containing $\\mathfrak{U}$ and $E$ a normal subgroup of $G$ with $G/E\\in \\mathfrak{F}$. 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