The Lyons group has no distance-transitive representation

Summary

Abstract

We show that the Lyons group Ly has no distance-transitive representation, and that the only faithful multiplicity-free permutation representations of Ly are those on the cosets of G₂(5) and of 3 M cL: 2.

One area which is still open in the classification of primitive distance-transitive graphs is the determination of the primitive distance-transitive representations of certain sporadic simple groups and their automorphism groups (see [1]). In this note we show that the Lyons group Ly ≅ Ant(Ly) has no distance-transitive representation. In the process, we find that the only faithful multiplicity-free permutation representations of Ly are those on the cosets of G₂(5) and of 3 M cL:2.

Let G be a permutation group on a finite set Ω, and Γ a connected graph with vertex set Ω. (Throughout this note all graphs are undirected, with no loops and no multiple edges.) We say that G acts distance-transitively on Γ if for each i = 0,…, diam(Γ), G is transitive on the set of ordered pairs of vertices at distance i in Γ. The graph Γ is called distance-transitive if Aut(Γ) acts distance-transitively on Γ. A distance-transitive representation (DTR)

of a group X is a faithful permutation representation such that ρ(X) acts distance-transitively on some connected graph with vertex set Ω.

Let X be a finite group, and ρ : X → Sym(ω) a DTR.