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A subgroup $A$ of a finite group $G$ is said to be $K$-$\\mathfrak{F}$-subnormal in $G$ if there is a subgroup chain $$A=A_{0} \\leq A_{1} \\leq \\cdots \\leq A_{n}=G$$ such that either $A_{i-1} \\trianglelefteq A_{i}$ or $A_{i}/(A_{i-1})_{A_{i}} \\in \\mathfrak{F}$ for all $i=1, \\ldots , n$. A formation $\\frak {F}$ is said to be $K$-lattice provided in every finite group $G$ the set of all its $K$-$\\mathfrak{F}$-subnormal subgroups forms a sublattice of the lattice of all subgroups of $G$.\n  In this paper we consider some new applications of the theory of $K$-latt"},"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":"1705.10476","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2017-05-30T07:09:02Z","cross_cats_sorted":[],"title_canon_sha256":"29aca0c40a183324fb115b15c473589d9bff36903380f82f1e3416869db2af2f","abstract_canon_sha256":"81085b3e0eeaaaf203440c524638b765278aacbe4fd4a76470cc3d06382a36f1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:43:24.476798Z","signature_b64":"42WmyaeF72lyC/R6gDrQ/J/xKMQIHMI2RizBcQ5zSauiL/HUlAcYGonFniYrlEJUAlKV36SJ4H/nnQx92P9DBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1bd842266d5834e43808a4d9aa2fba75ba481a09c406d0c3e413b09af5ab495b","last_reissued_at":"2026-05-18T00:43:24.476230Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:43:24.476230Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Finite groups with systems of $K$-$\\frak{F}$-subnormal subgroups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Alexander N. 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