# class equation theorem

Let $G$ be a group acting on a finite set $X$. Define the set of invariants in $X$ by the action of $G$ as $G_{X}=\{x\in X\quad\lvert\quad gx=x\quad\forall g\in G\}$. Then there are $H_{1},...,H_{r}$ subgroups of $G$ with $H_{i}\neq G\quad\forall i$ such that $\#X=\#G_{X}+\sum_{i=1}^{r}[G:H_{i}]$

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