A Character Condition for Quadruple Transitivity

Abstract

Let 𝐺 be a permutation group of degree 𝑛 viewed as a subgroup of the symmetric group 𝑆≅𝑆𝑛. We show that if the irreducible character of 𝑆 corresponding to the partition of 𝑛 into subsets of sizes 𝑛−2 and 2, that is, to say the character often denoted by 𝜒(𝑛−2,2), remains irreducible when restricted to 𝐺, then 𝑛 = 4, 5 or 9 and 𝐺≅𝑆3, A5, or PΣL2(8), respectively, or 𝐺 is 4-transitive.