{"paper":{"title":"Classification of finite groups with toroidal or projective-planar permutability graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.GR","authors_text":"Andrei Gagarin, P. Devi, R. Rajkumar","submitted_at":"2015-05-13T17:24:46Z","abstract_excerpt":"Let $G$ be a group. The permutability graph of subgroups of $G$, denoted by $\\Gamma(G)$, is a graph having all the proper subgroups of $G$ as its vertices, and two subgroups are adjacent in $\\Gamma(G)$ if and only if they permute. In this paper, we classify the finite groups whose permutability graphs are toroidal or projective-planar. In addition, we classify the finite groups whose permutability graph does not contain one of $K_{3,3}$, $K_{1,5}$, $C_6$, $P_5$, or $P_6$ as a subgraph."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.03462","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"}