Let G be a group of order 15 acting on a set of order 22. Assume there are no fixed points. Determine how many orbits there are.
2026-04-04 03:40:19.1775274019
Number of Orbits in Group Action
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The size of an orbit $\mathcal{O}_x$ of $x$ in our set $S$ is equal to the index of the stabilizer $G_x$ in $G$. Since no element is fixed, we cannot have an orbit of size $1$, and the only other possible indices are $3,5$, and $15$ by Lagrange's theorem.
So, somehow we need to figure out how some positive integral combination of $3$, $5$, and $15$ can sum to $22$. It turns out there is a unique way to do this. See if you can determine the size and number of the orbits from here.