Virtual braids and permutations

TL;DR

This paper classifies all homomorphisms between virtual braid groups and symmetric groups, revealing structural properties such as automorphism groups and Hopfian characteristics of virtual braid groups.

Contribution

It provides a complete classification of homomorphisms between VB$_n$ and symmetric groups, and establishes key algebraic properties of VB$_n$.

Findings

Out(VB$_n$) is isomorphic to Z/2Z x Z/2Z

VB$_n$ is Hopfian and co-Hopfian

Classified all homomorphisms between VB$_n$ and symmetric groups

Abstract

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.