Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Hurwitz Equivalence in Dihedral Groups. | Hurwitz Equivalence in Dihedral Groups Emily Berger Massachusetts Institute of Technology ERB90@ Submitted Nov 18 2009 Accepted Jan 25 2010 Published Feb 21 2011 Mathematics Subject Classification 20F36 20F55 Abstract 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 E 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 Gn be the cartesian product of G with itself n times. The braid group Bn acts on Gn by Hurwitz moves. We study the orbits of this action when G is a dihedral group. When the tuple T E Gn 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 m 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 2pa 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 .
