What group action has $\forall x,y,\exists g:g\cdot x=g\cdot y$?

Question

Consider the action of $G$ on $X$.

Let it be a property of $G,X$ that $\forall x,y,\exists g:g\cdot x=g\cdot y$. This is not quite a transitive action - it describes for example a sequence of inclusions. What is the name for this type of action? I can't pair it with an appropriate definition from here.

My attempt? There seem to be several things going on here, none of which I can associate with documented group theory at the moment.

$G$ seems to define a "contracting epimorphism"

$G$ seems to define the identity function on the trivial group having the powerset of $X$ as its element.

Answer 1

No group action (on a set with more than one element) satisfies this. If $g\cdot x=g\cdot y$, then $x=g^{-1}\cdot(g\cdot x)=g^{-1}\cdot(g\cdot y)=y$, so your condition implies $x=y$ for all $x,y\in X$.

Answer 2

Well, notice that $g^{-1}\cdot (g\cdot x) = (g^{-1}g)\cdot x = 1\cdot x = x$.

Thus $g\cdot x = g\cdot y$ implies by applying $g^{-1}$ that $x=y$.

Answer 3

This is only possibly if $X$ is a singleton (or, vacuously, if $X$ is empty). To see this: let $x, y \in X$ and suppose $g$ is such that $g \cdot x = g \cdot y$. Then $x = g^{-1}g \cdot x = g^{-1}g \cdot y = y$. So all elements in $X$ are equal, and thus $X$ is a singleton . It is not hard to see that if $X$ is a singleton, this property holds.

Since this property can only hold in uninteresting cases, I doubt it has a name.