On isomorphisms of finite Cayley graphs

Abstract

A Cayley graph Cay (G, S) of a group G is called a CI-graph if whenever T is another subset of G for which Cay(G, S) ≅ Cay(G, T), there exists an automorphism σ of G such that Sσ = T. For a positive integer m, the group G is said to have the m-CI property if all Cayley graphs of G of valency m are CI-graphs; further, if G has the k-CI property for all k ≤ m, then G is called an m-CI-group, and a |G|-CI-group G is called a CI-group. In this paper, we prove that A5 is not a 5-CI-group, that SL(2, 5) is not a 6-CI-group, and that all finite 6-CI-groups are soluble. Then we show that a nonabelian simple group has the 4-CI property if and only if it is A5, and that no nonabelian simple group has the 5-CI property. Also we give nine new examples of CI-groups of small order, which were found to be CI-groups with the assistance of a computer. © 1998 Academic Press.