Group of order 15 acting on a set of order 22 If a group of order 15 acts on a set of order 22 and there are no fixed points. How many orbits are there?
I know the group action corresponds to a homomorphism from G into $S_{22}$

Mariano's comment is the key point, but maybe you need more of a push. Let's call the group $G$ and the set $S$.
$S_{22}$ is a pretty large group, so my mind wouldn't jump there. Rather, the orbits of the action partition $S$, and if $s \in S$ then the size of the orbit $Gs$ containing $s$ is the index $(G : G_s)$ of the stabilizer subgroup of $s$. Using Lagrange's theorem and the fact that $G$ has no fixed points (What restriction does this place on the order of $G_s$?) we get three possibilities for the index.
Now this becomes a familiar problem: how many ways can you make $22$ using addition and these three numbers? There should be only one.
