alternative characterization of multiply transitive permutation groups

This article derives an alternative characterization of n-transitive groups.


For n>1, G is n-transitiveMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath on X if and only if for all xX, Gx is (n-1)-transitive on X-{x}.


First assume G is n-transitive on X, and choose xX. To show Gx is (n-1)-transitive on X-{x}, choose x1,,xn-1,y1,,yn-1X. Since G is n-transitive on X, we can choose σG such that


But obviously σGx, and σ restricted to X-{x} is the desired permutationMathworldPlanetmath.

To prove the converseMathworldPlanetmath, choose x1,,xn,y1,,ynX. Choose σ1Gxn such that


and choose σ2Gy1 such that


Then σ2σ1 is the desired permutation. ∎

Note that this definition of n-transitivity affords a straightforward proof of the statement that An is (n-2)-transitive: by inspection, A3 is 1-transitive; the result follows by inductionMathworldPlanetmath using the theorem. (The corresponding statement that Sn is n-transitive is obvious).

Finally, note that the most common cases of n-transitivity are for n=1 (transitive), and n=2 (doubly transitive).

Title alternative characterization of multiply transitive permutation groups
Canonical name AlternativeCharacterizationOfMultiplyTransitivePermutationGroups
Date of creation 2013-03-22 17:21:47
Last modified on 2013-03-22 17:21:47
Owner rm50 (10146)
Last modified by rm50 (10146)
Numerical id 4
Author rm50 (10146)
Entry type DerivationPlanetmathPlanetmath
Classification msc 20B20
Defines doubly transitive