# Number of orbits for the action of a group on a set

Problem: Find the number of orbits in the set $$X:=\{1,2,\dots, 10\}$$ for the action of the cyclic subgroup $$G=\langle(1,3,5,7)\rangle$$ of $$S_{10}$$.



Attempt: Plugging into the Orbit-Counting Theorem, I get $$\sum\limits_{x \in X} |\text{Stab}_G(x)|$$. There are $$6$$ elements in $$X$$ unaffected by $$(1,3,5,7)$$ so I conclude that there are $$6$$ orbits.



Question: Is my application of the Orbit-Counting Theorem correct?

$$\{1,3,5,7\}$$ is also an orbit, since the lone generator of $$G$$ can send any element to any element within this set, so there are 7 orbits, not 6.