having trouble with this problem. It's homework. We're given a finite set $S$ on which a finite group $G$ acts on transitively. If $U$ is a subset of $S$, I'm supposed to show that the subsets of $gU$ cover $S$ evenly. By evenly, I mean that each $s\in S$ is in the same number of sets $gU$.
Things I know are that for any g,g'\in G, $gU$ and g'U both have order $|U|$. I also know that, since the operation is transitive, there is only one orbit (and I suspect this is important).
I also noticed that, when $|U| = 1$, this property is just the transitivity of the group action. I'm just having trouble generalizing this to where the sets overlap (ie, $s\in S$ is in more than one set $gU$.)