# simply transitive

Let $G$ be a group acting on a set $X$. The action is said to be simply transitive if it is transitive and $\forall x,y\in X$ there is a unique $g\in G$ 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 $g=1_{G}$ because $1_{G}.x=x$ also. Now assume the action is free and transitive and we have elements $g_{1},g_{2}\in G$ and $x,y\in X$ such that $g_{1}.x=y$ and $g_{2}.x=y$. Then $g_{1}.x=g_{2}.x\implies g_{2}^{-1}.g_{1}.x=(g_{2}^{-1}g_{1}).x=x$ hence $g_{2}^{-1}g_{1}=1_{G}$ because the action is free. Thus $g_{1}=g_{2}$ and so the action is simply transitive. ∎

Title simply transitive SimplyTransitive 2013-03-22 14:37:41 2013-03-22 14:37:41 benjaminfjones (879) benjaminfjones (879) 7 benjaminfjones (879) Definition msc 20M30 GroupAction