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

TL;DR

This paper investigates finite non-abelian BI-groups that are not CI-groups, providing examples of such groups of orders 20 and 42, and listing all BI-groups up to order 30.

Contribution

It identifies the first known non-abelian BI-groups that are not CI-groups and catalogs BI-groups of small orders, advancing understanding of their properties.

Findings

Found non-abelian BI-groups of orders 20 and 42 that are not CI-groups.

Listed all BI-groups of order up to 30.

Confirmed that not all BI-groups are CI-groups, especially in the non-abelian case.

Abstract

Let $G$ be a group and $S$ an inverse closed subset of $G\setminus \{1\}$. 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 $x^{-1}y\in S$. A group $G$ is called a CI-group if $Cay(G,S)\cong Cay(G,T)$ for some inverse closed subsets $S$ and $T$ of $G\setminus \{1\}$, then $S^\alpha=T$ for some automorphism $\alpha$ of $G$. A finite group $G$ is called a BI-group if $Cay(G,S)\cong Cay(G,T)$ for some inverse closed subsets $S$ and $T$ of $G\setminus \{1\}$, then $M_\nu^S=M_\nu^T$ for all positive integers $\nu$, where $M_\nu^S$ denotes the set $\big\{\sum_{s\in S}\chi(s) | \chi(1)=\nu, \chi \text{ is a complex irreducible character of } G \big\}$. It was asked by L\'aszl\'o Babai [\textit{J. Combin. Theory Ser. B}, {\bf 27} (1979) 180-189] if every finite group is a BI-group; various examples of finite non BI-groups are presented in [\textit{Comm. Algebra}, {\bf 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 $30$.