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State/Counting Sylow subgroups
Theorem: (Counting Sylow subgroups) Suppose that $G$ is a finite group, and $p$ is a prime number. Let $n$ be the largest natural number, such that $p^n$ divides $G$. Let $m$ be the unique natural number, which satisfies: $$\vert G \vert = p^n \cdot m.$$ Let $s_p$ be the number of $p$-Sylow subgroups of $G$. Then, the following statements are true:
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