Transitive Group Action

up:: Group Action

A Group Action is said to be transitive if, for any two points in the base space $X$, there exists some element in the group which “connects” them.

That is, given a base space $X$ and a Group $G$ which acts upon $X$ (e.g. on the left),

$$\forall x,y\in X,\exists g\in G\mid y=g\cdot x$$

Corollary

Transitive Group Actions are Surjective, since for all points $y\in X$, $\exists (g_x,x=g_x^{-1}\cdot y)\in (G,X)$ such that $g_x\cdot x=y$.

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References

• Group action - Wikipedia