Let G be a group of order 15 action on a set X of cardinality 17. Suppose that for all x ∈ X, we have $|O_x| ≥ 2$. Determine the number of orbits and the cardinality of each orbit.
With a given lemma, I know that $|X| = \sum_{distinct orbits O_x} |O_x|$, so I get $17 =\sum_{distinct orbits O_x} |O_x|$, and because $|O_x| \geq 2$, I know that $|O_x| \in \{3,5,15 \}$. Where do I go from here?
The following equation must be solved for nonnegative integers $n,m,k$:$$3n+5m+15k=17$$
And it is almost immediate that $k=0$ is the only option here, so we get:$$3n+5m=17$$
Now just solve that by checking the possibilities one by one.