A transitive group action 
on a nonempty set 
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, there is a bijection 
to the set
of left cosets of the isotropy
group 
that commutes with the group action. If 
is a smooth manifold, 
is a Lie group, and the transitive group action is a
smooth map, then 
is a homogeneous space.
If 
has at least two elements and, 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 for a positive
integer 
if 
has at least 
elements and, for every two ordered k-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.