Transitive Group Action

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 gx=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 gx_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 gx_i=y_i.

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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Burnside, W. "On Transitive Groups of Degree n and Class n-1." 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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Transitive Group Action

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Rowland, Todd. "Transitive Group Action." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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