nLab
transitive action

Contents

Definition

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).

A transitive action that is also free is called regular.

Examples

  • Any group GG acts transitively on itself by multiplication :G×GG\cdot : G \times G \to G, which is called the (left) regular representation of GG.

  • 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 modular group PSL(2,)PSL(2,\mathbb{Z}) acts transitively on the rational projective line 1()={}\mathbb{P}^1(\mathbb{Q}) = \mathbb{Q} \cup \{\infty\}. The projective general linear group PGL(2,)PGL(2,\mathbb{C}) acts 3-transitively on the Riemann sphere 1()\mathbb{P}^1(\mathbb{C}).

As an action on cosets

Let ρ:G×XX\rho : G \times X \to X be a transitive action and suppose that XX is inhabited. Then ρ\rho is equivalent to the action of GG by multiplication on a coset space G/HG/H, where the subgroup HH is taken as the stabilizer subgroup

H=G x={gGgx=x}H = G_x = \{ g \in G \mid g x = x \}

of some arbitrary element xXx \in X. In particular, the transitivity of ρ\rho guarantees that the GG-equivariant map G/HXG/H \to X defined by gHgxg H \mapsto g x is a bijection. (Note that although the subgroup H=G xH = G_x depends on the choice of xx, it is determined up to conjugacy, and so the coset space G/HG/H is independent of the choice of element.)

References

Revised on May 2, 2016 10:22:38 by Urs Schreiber (131.220.184.222)