{"paper":{"title":"Non-Solvable Graph of a Finite Group and Solvabilizers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Doron Hai-Reuven","submitted_at":"2013-07-10T21:26:31Z","abstract_excerpt":"Let $G$ be a finite group. For $x \\in G$, we define the solvabilizer of $x$ in $G$, denoted $sol_G(x)$, to be the set $\\{g \\in G \\mid \\langle g,x \\rangle$ is solvable$\\}$. A group $G$ is an S-group if $sol_G(x)$ is a subgroup of $G$ for every $x \\in G$. In this paper we prove that $G$ is solvable $\\Leftrightarrow$ $G$ is an S-group. Secondly, we define the non-solvable graph of $G$ (denoted ${\\mathcal S}_{G}$). Its vertices are $G$ and there is an edge between $x,y \\in G$ whenever $\\langle x,y \\rangle$ is not solvable. If $S(G)$ is the solvable radical of $G$ and $G$ is not solvable, we look a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.2924","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"}