transitive action



An action

ρ:G×XX \rho : G \times X \to X

of a group GG on a set XX is transitive if it has a single orbit, i.e., if for every two points a,ba,b there exists gGg\in G such that b=ρ(g,a)b = \rho(g,a). A set equipped with a transitive action of GG (and which is inhabited) is the same thing as a connected object in the category GSetG Set.

For k0k\ge 0, an action G×XXG \times X \to X is said to be kk-transitive if the componentwise-action G×X k̲X k̲G \times X^{\underline{k}} \to X^{\underline{k}} is transitive, where X k̲X^{\underline{k}} denotes the set of tuples of kk distinct points (i.e., injective functions from {1,,k}\{1,\dots,k\} to XX). For instance, an action of GG on XX is 3-transitive if any pair of triples (a 1,a 2,a 3)(a_1,a_2,a_3) and (b 1,b 2,b 3)(b_1,b_2,b_3) of points in XX, where a ia ja_i \ne a_j and b ib jb_i \ne b_j for iji\ne j, there exists gGg \in G such that (b 1,b 2,b 3)=(ga 1,ga 2,ga 3)(b_1,b_2,b_3) = (g a_1,g a_2,g a_3).


Any group acts transitively on itself.

The alternating group A nA_n acts transitively on {1,,n}\{1,\dots,n\} for n>2n \gt 2, and in fact it acts (n2)(n-2)-transitively for all n2n \ge 2.

The projective general linear group PGL(2,)PGL(2,\mathbb{C}) acts 3-transitively on the Riemann sphere 1()\mathbb{P}^1(\mathbb{C}).


  • Helmut Wielandt. Finite Permutation Groups. Academic Press, 1964.

Revised on November 4, 2015 06:21:55 by Noam Zeilberger (