transitive action

An action

$\rho : G \times X \to X$

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

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 $(b_1,b_2,b_3) = (g a_1,g a_2,g a_3)$.

Any group acts transitively on itself.

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

- Helmut Wielandt.
*Finite Permutation Groups*. Academic Press, 1964.

Revised on November 4, 2015 06:21:55
by Noam Zeilberger
(193.55.177.48)