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单词 SimplyTransitive
释义

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.

Theorem.

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

Proof.

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.∎

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更新时间:2025/5/4 14:08:29