Transitive Group Action (original) (raw)
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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 
.
See also
Effective Action, Faithful Group Action, Free Action, Group,Group Orbit, Group Representation, Isotropy Group, Lie Group Quotient Space, Matrix Group, Topological Group, Transitive Group
This entry contributed by Todd Rowland
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References
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.
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Cite this as:
Rowland, Todd. "Transitive Group Action." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/TransitiveGroupAction.html