A group action is transitive if it possesses only a single group orbit, i.e., for every pair of elements and , there is a group element such that . In this case, is isomorphic to the left cosets of the isotropy group, . The space , 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 and , there is a group element such that , then the group action
is called doubly transitive. Similarly, a group action can be triply transitive and,
in general, a group action is -transitive if every set of distinct elements has a group element such that .
Burnside, W. "On Transitive Groups of Degree and Class ." Proc. London Math. Soc.32, 240-246,
1900.Hulpke, A. Konstruktion
transitiver Permutationsgruppen. Ph.D. thesis. Aachen, Germany: RWTH, 1996.
Also available as Aachener Beiträge zur Mathematik, No. 18, 1996.Kawakubo,
K. The
Theory of Transformation Groups. Oxford, England: Oxford University Press,
pp. 4-6 and 41-49, 1987.Rotman, J. Theory
of Groups. New York: Allyn and Bacon, pp. 180-184, 1984.
is transitive if it possesses only a single group orbit, i.e., for every pair of elements 
and 
, there is a group element 
such that 
. In this case, 
is isomorphic to the left cosets of the isotropy group, 
. The space 
, which has a transitive group action, is called a homogeneous
space when the group is a Lie group.
and 
, there is a group element 
such that 
, then the group action
is called doubly transitive. Similarly, a group action can be triply transitive and,
in general, a group action is 
-transitive if every set 
of 
distinct elements has a group element 
such that 
.
and Class 
." Proc. London Math. Soc. 32, 240-246,
1900.Hulpke, A.