A transitive group action is a group action that 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 ordered pairs and with distinct entries, there is a group element
such that
for ,
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, for every two ordered -tuples and whose entries are distinct within each tuple,
there is a group element such that for all .
Burnside, W. "On Transitive Groups of Degree and Class ." Proc. London Math. Soc.32, 240-246,
1900.Hulpke, A. Konstruktion
transitiver Permutationsgruppen. PhD 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 a group action that 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 
with distinct entries, there is a group element 
such that 
for 
,
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, for every two ordered 
-tuples 
and 
whose entries are distinct within each tuple,
there is a group element 
such that 
for all 
.
and Class 
." Proc. London Math. Soc. 32, 240-246,
1900.Hulpke, A.