A Character Condition for Quadruple Transitivity

LetG be a permutation group of degree n. The connection between themultiple transitivity of G and the irreducibility of certain irreducible characters of of the symmetric group Sn when restricted to G goes back to Frobenius, and a proof of his classical result can be found in Tsuzuku 1 . In 2 , Saxl strengthens this result, limiting the irreducibility hypothesis to the characters χ n−k,k G , a special case of his theorem having been proved by Neumann 3 . In 4 Saxl uses the Classification of Finite Simple Groups to gomuch further. He obtains a complete list of pairs χ,G where χ is an irreducible character of the symmetric group Sn, and G is a subgroup of Sn such that the restriction of χ to G remains irreducible. The present paper is concerned with the case when χ n−2,2 G is irreducible, proving that G is 4-transitive except in three small cases which are worked out explicitly. As such, the result, which appears as Theorem 3.3, can be deduced from 2, 3, 5 . However, the approach used here is self-contained and elementary.