Virtual braids and permutations

Abstract

Let VB_n be the virtual braid group on n strands and let S_n be the symmetric group on n letters. Let n, m ∈ N such that n ≥ 5, m ≥ 2 and n ≥ m. We determine all possible homomorphisms from VB_n to S_m , from S_n to VB_m and from VB_n to VB_m. As corollaries we get that Out(VB_n) is isomorphic to the Klein group and that VB_n is both Hopfian and co-Hofpian.