A group action G×X->X is transitive if it possesses only a single group orbit, i.e., for every pair of elements x and y, there is a group element g such that g x = y. In this case, X is isomorphic to the left cosets of the isotropy group, X~G/G_x. The space X, which has a transitive group action, is called a homogeneous space when the group is a Lie group. If, for every two pairs of points x_1, x_2 and y_1, y_2, there is a group element g such that g x_i = y_i, then the group action is called doubly transitive. Similarly, a group action can be triply transitive and, in general, a group action is k-transitive if every set {x_1, ..., y_k} of 2k distinct elements has a group element g such that g x_i = y_i.

effective action | faithful group action | free action | group | group orbit | group representation | isotropy group | Lie group quotient space | matrix group | topological group | transitive group