On Frobenius graphs of diameter \(3\) for finite groups (Q6668550)

Let \(G\) be a finite group and \(H \leq G\). The Frobenius graph \(\Gamma(G,H)\) is the bipartite graph with vertex set the disjoint union of \(\mathrm{Irr}(G)\) and \(\mathrm{Irr}(H)\) and an edge between \(\chi \in \mathrm{Irr}(G)\) and \(\rho \in \mathrm{Irr}(H)\) whenever \([\chi_{H},\rho]\not =0\). Let \(d\) be diameter of \(\Gamma(G,H)\), that is, the largest distance of two vertices in \(\Gamma(G,H)\), and assume that \(d\) is finite. Frobenius graphs can have arbitrarily large diameters, for example the diameter of \(\Gamma(S_{n+1}, S_{n})\) is \(2n\) (\(S_{n}\) is the symmetric group of degree \(n\)).\N\NIn the paper under review, the authors study the case \(d=3\) (in this case \(H\) is called a diameter three subgroup of \(G\)). In particular, among other things, they prove Theorem 5.1: Let \(G\) be a quasisimple group. Then \(G\) has a diameter three subgroup, except if \(G \simeq \mathrm{SL}(2, 5)\simeq 2.A_{5}\) or \(G \simeq \mathrm{SL}(2, 9) \simeq 2.A_{6}\) or \(G \simeq 6.A_{6}\).