{"paper":{"title":"Virtual braids and permutations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Luis Paris (IMB), Paolo Bellingeri (LMNO)","submitted_at":"2018-08-30T13:58:09Z","abstract_excerpt":"Let VB$_n$ be the virtual braid group on $n$ strands and let $\\mathfrak{S}_n$ be the symmetric group on $n$ letters. Let $n,m \\in \\mathbb{N}$ such that $n \\ge 5$, $m \\ge 2$ and $n \\ge m$. We determine all possible homomorphisms from VB$_n$ to $\\mathfrak{S}_m$, from  $\\mathfrak{S}_n$ to VB$_m$ and  from  VB$_n$ to VB$_m$. As corollaries we get that Out(VB$_n$) is isomorphic to $\\mathbb{Z}/2\\mathbb{Z} \\times \\mathbb{Z}/2\\mathbb{Z}$ and that VB$_n$ is both Hopfian and co-Hofpian."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.10301","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"}