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‎We call a group satisfying the following property for all $\nu \in cd(G)$ (Irreducible character degrees of $G$) a BI-group (Babai Invariant group)

Let $G$ be a finite group‎, ‎let $\Gamma=Cay(G,S)$ be a Cayley graph‎, ‎$\nu\in cd(G)$ (an irreducible character degree of $G$). Then $M_\nu^S=\{\sum_{s\in S}\chi_i(s) \;\vert\; \chi_i(1)=\nu, \chi_i \in Irr(G)\}$ is an invariant of $\Gamma$. (Thus‎, $Cay(G,S)\cong Cay(G,T)$ implies that $M_\nu^S=M_\nu^T$).

Now, I have two questions:

1) Is $A_5$ a BI-group?

2) Which non-abelian groups of order less than $100$ are BI-group?

We do it by GAP [1], but it is very time consuming and infeasible.

[1] A. Abdollahi and M. Zallaghi, MR 3395696 Character Sums for Cayley Graphs, Comm. Algebra 43 (2015), no. 12, 5159--5167.

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  • $\begingroup$ Maybe by computation with MAGMA and by transforming the memory domain to time domain in programming, we can study all cases. The question seems interesting. $\endgroup$
    – Shahrooz
    Commented Dec 14, 2016 at 4:27

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