# What's a Transitive Group Action?

Let a group $G$ act on a set $X$. The action is said to be transitive if for any two $x,y\in X$ there is a $g\in G$ such that $g\cdot x=y$. This is equivalent to saying there is an $x\in X$ such that $\text{orb}(x)=X$, i.e. there is exactly one orbit. And all this is just the fancy way of saying that $G$ shuffles all the elements of $X$ among themselves. In other words,

"What happens in $X$ stays in $X$."

Illustration: Imagine that $X$ is a little box of marbles resting on the floor of a room $Y$, and let $G$ act on $Y$ so that all the elements inside the room (including the marbles in the box $X$!) get tossed around. (So $G$ is like an earthquake, or something....) If the action of $G$ is transitive, then none of the marbles in the little box fell outside the box and onto the floor ($Y$). Even though the elements of $X$ were shuffled, they remained within $X$.

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