nLab transitive action

Definition

An action

$$(-)\cdot(-) \;\colon\; G \times X \to X$$

of a group $G$ on a set $X$ is called transitive if it has a single orbit, i.e. for any two elements, $x, y \in X$, there exists $g\in G$ such that $y = g \cdot x$.

This is equivalent to saying that the shear map

$$\array{ G \times X & \overset { (pr_2, \cdot) } {\longrightarrow}& X \times X \\ (g, x) & \mapsto & \big( x, g \cdot x \big) \,. }$$

is an epimorphism. In this form the definition makes sense for action objects internal to any ambient category with finite products (where one may want to require regular, effective, split, … epimorphisms, all of which notions coincide in the context of Sets).

Beware that often it is assumed that the underlying object $X$ of a transitive action is inhabited (but not always, see at pseudo-torsor).

For $k\ge 0$, an action $G \times X \to X$ is said to be $k$-transitive if the componentwise-action $G \times X^{\underline{k}} \to X^{\underline{k}}$ is transitive, where $X^{\underline{k}}$ denotes the set of tuples of $k$ distinct points (i.e., injective functions from $\{1,\dots,k\}$ to $X$). For instance, an action of $G$ on $X$ is 3-transitive if any pair of triples $(a_1,a_2,a_3)$ and $(b_1,b_2,b_3)$ of points in $X$, where $a_i \ne a_j$ and $b_i \ne b_j$ for $i\ne j$, there exists $g \in G$ such that $(b_1,b_2,b_3) = (g a_1,g a_2,g a_3)$.

A transitive action that is also free is called regular action. See also at torsor.

Properties

A set equipped with a transitive action of $G$ (and which is inhabited) is the same thing as a connected object in the category of G-sets. A $G$-set may be decomposed uniquely as a coproduct of transitive $G$-sets.

Examples

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

• The alternating group $A_n$ acts transitively on $\{1,\dots,n\}$ for $n \gt 2$, and in fact it acts $(n-2)$-transitively for all $n \ge 2$.

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

• An action of $\mathbb{Z}$ (viewed as the free group on one generator) on a set $X$ corresponds to an arbitrary permutation $\pi : X \to X$, but the action is transitive just in case $\pi$ is a cyclic permutation.

As an action on cosets

Let $* : G \times X \to X$ be a transitive action and suppose that $X$ is inhabited. Then $*$ is equivalent to the action of $G$ by multiplication on a coset space $G/H$, where the subgroup $H$ is taken as the stabilizer subgroup

$$H = G_x = \{ g \in G \mid g * x = x \}$$

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

Related concepts

• combinatorial map

• homogeneous space

• torsor

• transitive action, free action, semi-free action, regular action

• proper action, properly discontinuous action

References

• Helmut Wielandt, Finite Permutation Groups, Academic Press (1964) [ark:/13960/t49q0xd7v]

• Group Properties Wiki, Fundamental theorem of group actions