of a group on a set is transitive if it has a single orbit, i.e., (i) is inhabited and (ii) for any two elements, , there exists such that .
This definition rules out the action of on the empty set. Note that transitivity of an action is sometimes defined via (ii) alone.
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 .
A set equipped with a transitive action of (and which is inhabited) is the same thing as a connected object in the category of G-sets. A -set may be decomposed uniquely as a coproduct of transitive -sets.
Any group acts transitively on itself by multiplication , which is called the (left) regular representation of .
The alternating group acts transitively on for , and in fact it acts -transitively for all .
of some arbitrary element . In particular, the transitivity of guarantees that the -equivariant map defined by is a bijection. (Note that although the subgroup depends on the choice of , it is determined up to conjugacy, and so the coset space 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.