proof of class equation theorem

$X$ is a finite disjoint union of finite orbits: $X=\\cup_{i}Gx_{i}$ . We can separate this union by considerating first only the orbits of 1 element and then the rest: $X=\\cup_{j=1}^{l}\\{x_{i_{j}}\\}\\cup\\cup_{k=1}^{s}Gx_{i_{k}}=G_{X}\\cup_{k=1}^{s}%Gx_{i_{k}}$ Then using the orbit-stabilizer theorem, we have $\\#X=\\#G_{X}+\\sum_{k=1}^{s}[G:G_{x_{i_{k}}}]$ where for every $k$ , $[G:G_{x_{i_{k}}}]\\geq 2$ , because if one of them were 1, then it would be associated to an orbit of 1 element, but we counted those orbits first. Then this stabilizers are not $G$ . This finishes the proof.

单词	ProofOfClassEquationTheorem
释义	proof of class equation theorem $X$ is a finite disjoint union of finite orbits: $X=\\cup_{i}Gx_{i}$ . We can separate this union by considerating first only the orbits of 1 element and then the rest: $X=\\cup_{j=1}^{l}\\{x_{i_{j}}\\}\\cup\\cup_{k=1}^{s}Gx_{i_{k}}=G_{X}\\cup_{k=1}^{s}%Gx_{i_{k}}$ Then using the orbit-stabilizer theorem, we have $\\#X=\\#G_{X}+\\sum_{k=1}^{s}[G:G_{x_{i_{k}}}]$ where for every $k$ , $[G:G_{x_{i_{k}}}]\\geq 2$ , because if one of them were 1, then it would be associated to an orbit of 1 element, but we counted those orbits first. Then this stabilizers are not $G$ . This finishes the proof.
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