{"paper":{"title":"On the lattice of the $\\sigma$-permutable subgroups of a finite group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Alexander N. Skiba","submitted_at":"2017-03-06T09:15:34Z","abstract_excerpt":"Let $\\sigma =\\{\\sigma_{i} | i\\in I\\}$ be some partition of the set of all primes $\\Bbb{P}$, $G$ a finite group and $\\sigma (G) =\\{\\sigma_{i} |\\sigma_{i}\\cap \\pi (G)\\ne \\emptyset \\}$. A set ${\\cal H}$ of subgroups of $G$ is said to be a complete Hall $\\sigma $-set of $G$ if every member $\\ne 1$ of ${\\cal H}$ is a Hall $\\sigma_{i}$-subgroup of $G$ for some $\\sigma_{i}\\in \\sigma $ and ${\\cal H}$ contains exactly one Hall $\\sigma_{i}$-subgroup of $G$ for every $\\sigma_{i}\\in \\sigma (G)$. A subgroup $A$ of $G$ is said to be ${\\sigma}$-permutable in $G$ if $G$ possesses a complete Hall $\\sigma $-set"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.01773","kind":"arxiv","version":2},"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"}