Contents

Definition

An action

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

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

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