Non-abelian finite groups whose character sums are invariant but are not cayley isomorphism

Abstract

Let G be a group and S an inverse closed subset of G\{1}. By a Cayley graph Cay(G,S), we mean the graph whose vertex set is the set of elements of G and two vertices x and y are adjacent if x-1y S. A group G is called a CI-group if Cay(G,S)≅Cay(G,T) for some inverse closed subsets S and T of G\{1}, then Sα = T for some automorphism α of G. A finite group G is called a BI-group if Cay(G,S)≅Cay(G,T) for some inverse closed subsets S and T of G\{1}, then M V S = M V T for all positive integers V, where M V S denotes the set {ΣsϵSX(s)|X(1) = V,X is a complex irreducible character of G}. It was asked by László Babai [Spectra of Cayley graphs, J. Combin. Theory Ser. B 27 (1979) 180-189] if every finite group is a BI-group; various examples of finite non-BI-groups are presented in [A. Abdollahi and M. Zallaghi, Character sums of Cayley graph, Comm. Algebra 43(12) (2015) 5159-5167]. It is noted in the latter paper that every finite CI-group is a BI-group and all abelian finite groups are BI-groups. However, it is known that there are finite abelian non-CI-groups. Existence of a finite non-abelian BI-group which is not a CI-group is the main question which we study here. We find two non-abelian BI-groups of orders 20 and 42 which are not CI-groups. We also list all BI-groups of orders up to 30. © 2019 World Scientific Publishing Company.