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A subgroup H of G is said to be {\\sigma}-permutable if G possesses a complete Hall {\\sigma}-set \\mathcal{H} such that HA^x = A^xH for all A \\in \\mathcal{H} and all x \\in G. We say that a subgroup H of G is weakly {\\sigma}-permutable in G if there exists a {\\sigma}-subnormal subgroup T of G such that"},"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":"1608.03224","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-08-10T16:11:33Z","cross_cats_sorted":[],"title_canon_sha256":"e4173af2243b67267de723bcd03a1479c5eb05eb006ba1a7222c29c633ffe68e","abstract_canon_sha256":"f615a524f3220a45ae23586b9ac1c876b8cc026534061ae30d41231d7b3f8886"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:09:29.985763Z","signature_b64":"/35H//zVhalbV5J6B6+e4O+WoW1GInxbxnoWiIp5XQlVyh59YWyQsMh3RfzuVBmvORIz+ojn21aayiFhw9B6AA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"47864e937c9de9ee84a3702b26bbd66678130e8dc6c07fa6bd8482cc6ec1d145","last_reissued_at":"2026-05-18T01:09:29.985389Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:09:29.985389Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On weakly $sigma$-permutable subgroups of finite groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Chi Zhang, W. Guo, Zhenfeng Wu","submitted_at":"2016-08-10T16:11:33Z","abstract_excerpt":"Let G be a finite group and {\\sigma} = {{\\sigma}_i, i \\in I} be a partition of the set of all primes \\mathbb{P}. A set \\mathcal{H} of subgroups of G with 1 \\in \\mathcal{H} is said to be a complete Hall {\\sigma}-set of G if every non-identity member of \\mathcal{H} is a Hall {\\sigma}_i-subgroup of G. A subgroup H of G is said to be {\\sigma}-permutable if G possesses a complete Hall {\\sigma}-set \\mathcal{H} such that HA^x = A^xH for all A \\in \\mathcal{H} and all x \\in G. 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