{"paper":{"title":"A note on the $R_\\infty$ property for groups $\\mathrm{FAlt}(X)\\leqslant G\\leqslant \\mathrm{Sym}(X)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Charles Cox","submitted_at":"2016-02-08T18:37:51Z","abstract_excerpt":"Given a set $X$, the group $\\mathrm{Sym}(X)$ consists of all bijections from $X$ to $X$, and $\\mathrm{FSym}(X)$ is the subgroup of maps with finite support i.e. those that move only finitely many points in $X$. We describe the automorphism structure of groups $\\mathrm{FSym}(X)\\le G\\le \\mathrm{Sym}(X)$ and use this to state some conditions on $G$ for it to have the $R_\\infty$ property. Our main results are that if $G$ is infinite, torsion, and $\\mathrm{FSym}(X)\\le G\\le \\mathrm{Sym}(X)$, then it has the $R_\\infty$ property. Also, if $G$ is infinite and residually finite, then there is a set $X$ "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.02688","kind":"arxiv","version":3},"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"}