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 $mathbb{Z}/2mathbb{Z} times mathbb{Z}/2mathbb{Z}$ and that VB$_n$ is both Hopfian and co-Hofpian.
تحميل البحث