Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Hurwitz Equivalence in Tuples of Generalized Quaternion Groups and Dihedral Groups. | Hurwitz Equivalence in Tuples of Generalized Quaternion Groups and Dihedral Groups Xiang-dong Hou Department of Mathematics University of South Florida Tampa FL 33620 xhou@ Submitted Apr 6 2008 Accepted May 29 2008 Published Jun 13 2008 Mathematics Subject Classifications 20F36 20F05 Abstract Let Q2m be the generalized quaternion group of order 2m and Dn the dihedral group of order 2N. We classify the orbits in Qnm and 1 2 p prime under the Hurwitz action. 1 The Hurwitz Action Let G be a group. For a b 2 G let ab b 1 ab and ba bab 1. The Hurwitz action on Gn n 2 is an action of the n-string braid group Bn on Gn. Recall that Bn is given by the presentation Bn hơ1 . ơn 1 I ơiơj ơjơi i - jI 2 ơiơi 1 ơi ơi 1ơiơi 1 1 i n - 2 . The action of ơi on Gn is defined by ơi ai . an ai . a i-1 ai 1 a i 1 ai 2 . an where a1 . an 2 Gn. Note that ơ. 1 a1 . an a1 . ai-1 aiai 1 ai ai 2 . an . An action by ơi or ơ. 1 on Gn is called a Hurwitz move. Two tuples a1 . an b1 . bn 2 Gn are called Hurwitz equivalent denoted as a1 . an b1 . bn if they are in the same Bn-orbit. The Hurwitz equivalence class of a1 . an 2 Gn . the Bn-orbit of a1 . an is denoted by a1 . an . If G is a nonabelian group in general the Bn-orbits in Gn are not known. In 1 Ben-Itzhak and Teicher determined all Bn-orbits in sm represented by t1 . tn where Sm is the symmetric group each ti is a transposition and t1 tn 1. It is obvious that if THE ELECTRONIC JOURNAL OF COMBINATORICS 15 2008 R80 1 a1 . an 2 G generate a finite subgroup then the Bn-orbit of a15. an in Gn is finite. It has been proved that if s1 . sn 2 GL Rn are reflections such that the Bn-orbit of s1 . sn is finite then the group generated by s1 . sn is finite see 2 and 3 . It is natural to ask which types of nonabelian group G allow complete determination of the Bn-orbits in Gn. In this paper we show that when G is the generalized quaternion group Q2m or the dihedral group Dpm of order 2pm where p is a prime the answer to the above .
