{"paper":{"title":"Commutativity pattern of finite non-abelian $p$-groups determine their orders","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.GR","authors_text":"A. Abdollahi, H. Dorbidi, H. Shahverdi, S. Akbari","submitted_at":"2011-09-23T06:48:35Z","abstract_excerpt":"Let $G$ be a non-abelian group and $Z(G)$ be the center of $G$. Associate a graph $\\Gamma_G$ (called non-commuting graph of $G$) with $G$ as follows: take $G\\setminus Z(G)$ as the vertices of $\\Gamma_G$ and join two distinct vertices $x$ and $y$, whenever $xy\\neq yx$. Here, we prove that \"the commutativity pattern of a finite non-abelian $p$-group determine its order among the class of groups\"; this means that if $P$ is a finite non-abelian $p$-group such that $\\Gamma_P\\cong \\Gamma_H$ for some group $H$, then $|P|=|H|$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.5007","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"}