this is a pretty basic question but I am struggling to make sense of the definitions. In Iwaniec's book on automorphic forms, page 29 he begins
Let $X$ be a topological space (Hausdorff) and $\Gamma$ a group of homeomorphisms of $X$ acting on $X$ (transitively). We say $\Gamma$ on $X$ discontinuously if the orbit $$\Gamma x=\{\gamma x:\gamma\in\Gamma\}$$ has no limit point in $X$.
This right here already has me confused as if $\Gamma$ acts transitively then $\Gamma x=X$ for any $x\in X$. What am I misunderstanding? I am guessing something else is meant by transitively but what would it be?
Thank you
