tailieunhanh - Báo cáo toán học: "On the Proof of a Theorem of P´lfy a"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: On the Proof of a Theorem of P´lfy a. | On the Proof of a Theorem of Palfy Edward Dobson Department of Mathematics and Statistics Mississippi State University PO Drawer MA Mississippi State MS 39762 USA dobson@ Submitted Mar 24 2006 Accepted Oct 10 2006 Published Oct 19 2006 Mathematics Subject Classification 05E99 Abstract Palfy proved that a group G is a CI-group if and only if G n where either gcd n n 1 or n 4 where is Euler s phi function. We simplify the proof of if gcd n n 1 and G is a group of order n then G is a CI-group . In 1987 Palfy 6 proved perhaps the most well-known result pertaining to the Cayley isomorphism problem. Namely that a group G of order n is a CI-group if and only if either gcd n n 1 or n 4 where is Euler s phi function. It is worth noting that every group of order n is cyclic if and only if gcd n n 1. It is the purpose of this note to simplify some parts of Palfy s original proof. Definition 1 Let G be a group and define gL G G by gL x gx. Let GL gL g 2 Gg. Then GL is the left-regular representation of G. It is a subgroup of the symmetric group Sg of all permutations on G. We define a Cayley object of G to be a combinatorial object X . digraph graph design code such that GL Aut X where Aut X is the automorphism group of X note that this implies that the vertex set of X is in fact G . To say that G is a CI-group means that if X and Y are any Cayley objects of G such that X is isomorphic to Y then some group automorphism of G is an isomorphism from X to Y . CI-groups are characterized by the following result due to Babai 1 . Lemma 1 For a group G the following are equivalent 1. G is a CI-group 2. for every 7 2 Sg there exists Ỗ 2 Gl 7-1GL7i such that J-17 1GL7Ổ GL. THE ELECTRONIC JOURNAL OF COMBINATORICS 13 2006 N16 1 We will not simplify all of Palfy s proof so it will be worthwhile to discuss exactly which part of his proof we will simplify. First we will not deal with groups G such that G 4 at all. Second we will only be concerned with showing that if gcd n