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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 $S_{10}$.

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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.

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

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$\{1,3,5,7\}$ 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.

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