of a group on a set is transitive if it has a single orbit, i.e., if for every two points there exists such that . A set equipped with a transitive action of (and which is inhabited) is the same thing as a connected object in the category .
For , an action is said to be -transitive if the componentwise-action is transitive, where denotes the set of tuples of distinct points (i.e., injective functions from to ). For instance, an action of on is 3-transitive if any pair of triples and of points in , where and for , there exists such that .
Any group acts transitively on itself.
The alternating group acts transitively on for , and in fact it acts -transitively for all .
The projective general linear group acts 3-transitively on the Riemann sphere .
- Helmut Wielandt. Finite Permutation Groups. Academic Press, 1964.