{"paper":{"title":"On $\\Pi$-permutable subgroups of finite groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"A.N.Skiba, Wenbin Guo","submitted_at":"2016-06-10T06:08:57Z","abstract_excerpt":"Let $\\sigma =\\{\\sigma_{i} | i\\in I\\}$ be some partition of the set of all primes $\\Bbb{P}$ and $\\Pi$ a non-empty subset of the set $\\sigma$. A set ${\\cal H}$ of subgroups of a finite group $G$ is said to be a \\emph{ complete Hall $\\Pi $-set} of $G$ if every member of ${\\cal H}$ is a Hall $\\sigma_{i}$-subgroup of $G$ for some $\\sigma_{i}\\in \\Pi$ and ${\\cal H}$ contains exact one Hall $\\sigma_{i}$-subgroup of $G$ for every $\\sigma_{i}\\in \\Pi$ such that $\\sigma_i\\cap \\pi(G)\\neq\\emptyset$. A subgroup $H$ of $G$ is called \\emph{$\\Pi$-quasinormal} or \\emph{$\\Pi$-permutable} in $G$ if $G$ possesses a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.03197","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}