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State/Counting Sylow subgroups
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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:
- $s_p$ divides $m$.
- $s_p$ is congruent to $1$, mod $p$.
- If $H$ is a $p$-Sylow subgroup, then $s_p = [G : N_G(H)]$ (the index of the normalizer of $H$ in $G$).
Logical Connections
This statement logically relies on the following definitions and statements: Def/Sylow subgroup, Def/Divides, Def/Congruent, Def/Coset, Def/Normalizer
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