Babai's problem

Unsolved problem in mathematics: Which finite groups are BI-groups? (more unsolved problems in mathematics)

Babai's problem is a problem in algebraic graph theory first proposed in 1979 by László Babai.

Babai's problem

Let [math]\displaystyle{ G }[/math] be a finite group, let [math]\displaystyle{ \operatorname{Irr}(G) }[/math] be the set of all irreducible characters of [math]\displaystyle{ G }[/math], let [math]\displaystyle{ \Gamma=\operatorname{Cay}(G,S) }[/math] be the Cayley graph (or directed Cayley graph) corresponding to a generating subset [math]\displaystyle{ S }[/math] of [math]\displaystyle{ G\setminus \{1\} }[/math], and let [math]\displaystyle{ \nu }[/math] be a positive integer. Is the set

[math]\displaystyle{ M_\nu^S=\left\{\sum_{s\in S} \chi(s)\;|\; \chi\in \operatorname{Irr}(G),\; \chi(1)=\nu \right\} }[/math]

an invariant of the graph [math]\displaystyle{ \Gamma }[/math]? In other words, does [math]\displaystyle{ \operatorname{Cay}(G,S)\cong \operatorname{Cay}(G,S') }[/math] imply that [math]\displaystyle{ M_\nu^S=M_\nu^{S'} }[/math]?

BI-group

A finite group [math]\displaystyle{ G }[/math] is called a BI-group (Babai Invariant group) if [math]\displaystyle{ \operatorname{Cay}(G,S)\cong \operatorname{Cay}(G,T) }[/math] for some inverse closed subsets [math]\displaystyle{ S }[/math] and [math]\displaystyle{ T }[/math] of [math]\displaystyle{ G\setminus \{1\} }[/math] implies that [math]\displaystyle{ M_\nu^S=M_\nu^T }[/math] for all positive integers [math]\displaystyle{ \nu }[/math].

Open problem

Which finite groups are BI-groups?

See also

• List of unsolved problems in mathematics
• List of problems solved since 1995