I am just trying to learn what transitive group action is. It seems I understand the definition, but I still have an issue.
We read on Wikipedia that (emphasis added): "The group action is transitive if and only if it has only one orbit, i.e. if there exists x in X with G.x = X. This is the case if and only if G.x = X for all x in X".
Direct implication of another definition I have (from the source I learn from) is that group action is transitive if for every x in X : G.x = X. According to Wikipedia (and later on in my source), it seems to be enough only for one x to have G.x = X to deduce transitivity.
In short, I don't see how G.x = X for one x in X => G.x = X for every x in X.
P.S: I have yet to learn what orbit is. So please avoid using orbits to explain the issue.