Hurwitz Equivalence in Dihedral Groups

Abstract

In this paper we determine the orbits of the braid group $B_n$ action on $G^n$ when $G$ is a dihedral group and for any $T \in G^n$. We prove that the following invariants serve as necessary and sufficient conditions for Hurwitz equivalence. They are: the product of its entries, the subgroup generated by its entries, and the number of times each conjugacy class (in the subgroup generated by its entries) is represented in $T$.