Hurwitz Equivalence in Dihedral Groups

In this paper we determine the orbits of the braid group Bn action on Gn when G is a dihedral group and for any T ∈ Gn. 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 . Introduction Let G be a group and G be the cartesian product of G with itself n times. The braid group Bn acts on G n by Hurwitz moves. We study the orbits of this action when G is a dihedral group. When the tuple T ∈ G consists only of reflections, the orbits are determined by the following invariants: the product of the entries, the subgroup generated by the entries, and the number of times each conjugacy class (in the subgroup generated by its entries) is represented in T . Our study of Hurwitz equivalence in the dihedral group was inspired by the paper [1], which gives a simple criterion for Hurwitz equivalence in the symmetric group analogous to our Main Theorem. That paper studies tuples of transpositions in the symmetric group, which is the reason why we originally chose to restrict to reflections in the dihedral group. (Recall that the symmetric group Sm acts on Rm−1 in such a way that every transposition acts by a Euclidean reflection.) Utlimately, we extend these results to include rotations as well. After the bulk of this work was completed we discovered the paper [3] that considers, using a different method, the case of a dihedral group of order 2p where p is prime. Our results were obtained independently and cover the case of dihedral groups of any order. In addition, after this paper was finished, [5] was published, extending the results of [3]. The results of our paper are complementary to the work in [5], since our results are derived from first principles using what is perhaps a more intuitive approach. the electronic journal of combinatorics 18 (2011), #P45 1