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An action
of a group on a set is called transitive if it has a single orbit, i.e. for any two elements, , there exists such that .
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 of a transitive action is inhabited (but not always, see at pseudo-torsor).
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 transitive action that is also free is called regular action. See also at torsor.
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 .
The modular group acts transitively on the rational projective line . The projective general linear group acts 3-transitively on the Riemann sphere .
An action of (viewed as the free group on one generator) on a set corresponds to an arbitrary permutation , but the action is transitive just in case is a cyclic permutation.
Let be a transitive action and suppose that is inhabited. Then is equivalent to the action of by multiplication on a coset space , where the subgroup is taken as the stabilizer subgroup
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) [ark:/13960/t49q0xd7v]
Last revised on May 12, 2024 at 18:03:14. See the history of this page for a list of all contributions to it.