Babai's problem

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $G$ be a finite group and $S=S^{-1}\subseteq G\setminus\{1\}$. The Cayley graph $\operatorname{Cay}(G,S)$ has vertex set $G$, with distinct vertices $x,y$ adjacent when $x^{-1}y\in S$. Let $\operatorname{Irr}(G)$ be the irreducible complex characters of $G$, and, for each $\nu\geq1$, put $M_\nu^S=\{\sum_{s\in S}\chi(s):\chi\in\operatorname{Irr}(G),\ \chi(1)=\nu\}$. These are sets, with no multiplicity. A BI-group is a finite group for which graph isomorphism $\operatorname{Cay}(G,S)\cong\operatorname{Cay}(G,T)$ implies $M_\nu^S=M_\nu^T$ for all $\nu\geq1$ and all inverse-closed $S,T\subseteq G\setminus\{1\}$; the subsets need not generate $G$.

Statement recorded in the supplied source.

Classify the finite BI-groups. [S2]

Short English statement

Which finite groups always preserve their sets of character sums of each character degree when two of their Cayley graphs are isomorphic?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 7400001 (AMR-073-0001). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Alireza Abdollahi and Maysam Zallaghi, Non-Abelian Finite Groups Whose Character Sums Are Invariant but Are Not Cayley Isomorphism (2017), v1; arXiv:1710.04446. https://arxiv.org/pdf/1710.04446.