simply transitive

Let G be a group acting on a set X. The action is said to be simply transitiveMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath if it is transitive and x,yX there is a unique gG such that g.x=y.


A group action is simply transitive if and only if it is free and transitive


Necessity follows since g.x=x implies that g=1G because 1G.x=x also. Now assume the action is free and transitive and we have elements g1,g2G and x,yX such that g1.x=y and g2.x=y. Then g1.x=g2.xg2-1.g1.x=(g2-1g1).x=x hence g2-1g1=1G because the action is free. Thus g1=g2 and so the action is simply transitive. ∎

Title simply transitive
Canonical name SimplyTransitive
Date of creation 2013-03-22 14:37:41
Last modified on 2013-03-22 14:37:41
Owner benjaminfjones (879)
Last modified by benjaminfjones (879)
Numerical id 7
Author benjaminfjones (879)
Entry type Definition
Classification msc 20M30
Related topic GroupAction