Transitive Group Actions are Surjective

up:: Transitive Group Action

Let $X$ be the base space, and $G$ a Group. Let $\alpha :G\times X\to X$ be a Transitive Group Action.

Let $y\in X$. We can know the inverse image of this group action, ${\alpha }^{-1}(y)$, as follows:
Since it is a transitive action, then $\forall x,y\in X,\exists g_x\in G\mid y=g_x\cdot x$. Acting on the left by $g_x^{-1}$ yields that

$$g_x^{-1}\cdot y=g_x^{-1}\cdot (g_x\cdot x)=(g_x^{-1}\cdot g_x)\cdot x=x$$

(the two last equalities follow from identity and compatibility of a group action.)

Thus, a transitive group action is surjective.