An action
of a group $G$ on a set $X$ is called transitive if it has a single orbit, i.e.,
$X$ is inhabited
for any two elements, $x, y \in X$, there exists $g\in G$ such that $y = g * x$.
(This definition rules out the action of $G$ on the empty set. Note that transitivity of an action is sometimes defined via (ii) alone.)
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.
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.
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.
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.)
Helmut Wielandt. Finite Permutation Groups. Academic Press, 1964.
Fundamental theorem of group actions on the Group Properties Wiki.
Last revised on April 24, 2018 at 09:17:45. See the history of this page for a list of all contributions to it.