Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: Some non-normal Cayley digraphs of the generalized quaternion group of certain orders | Some non-normal Cayley digraphs of the generalized quaternion group of certain orders Edward Dobson Department of Mathematics and Statistics PO Drawer MA Mississippi State MS 39762 . dobson@ Submitted Mar 10 2003 Accepted Jul 30 2003 Published Sep 8 2003 MR Subject Classifications 05C25 20B25 Abstract We show that an action of SL 2 p p 7 an odd prime such that 4 p 1 has exactly two orbital digraphs Pl r2 such that Aut Pj admits a complete block system B of p 1 blocks of size 2 i 1 2 with the following properties the action of Aut Pj on the blocks of B is nonsolvable doubly-transitive but not a symmetric group and the subgroup of Aut Pj that fixes each block of B set-wise is semiregular of order 2. If p 2k 1 7 is a Mersenne prime these digraphs are also Cayley digraphs of the generalized quaternion group of order 2k 1. In this case these digraphs are non-normal Cayley digraphs of the generalized quaternion group of order 2k 1. There are a variety of problems on vertex-transitive digraphs where a natural approach is to proceed by induction on the number of not necessarily distinct prime factors of the order of the graph. For example the Cayley isomorphism problem see 6 is one such problem as well as determining the full automorphism group of a vertex-transitive digraph r. Many such arguments begin by finding a complete block system B of Aut r . Ideally one would then apply the induction hypothesis to the groups Aut r B and fixAut r B b where Aut r B is the permutation group induced by the action of Aut r on B and fixAut r B is the subgroup of Aut r that fixes each block of B set-wise and B E B. Unfortunately neither Aut r B nor fixAut r B y need be the automorphism group of a digraph. In fact there are examples of vertex-transitive graphs where Aut r B is a doubly-transitive nonsolvable group that is not a symmetric group see 7 as well as examples of vertex-transitive graphs where fixAut r B b is a doubly-transitive nonsolvable group that is not
