Sylow theorems

Definition. Let $G$ be a finite group. For any prime $p$ , the subgroup with the highest power of $p$ is called the $p$ -Sylow subgroup of $G$ .

For example. For a group of order 120. The 2-Sylow subgroup has order 8, the 3-Sylow subgroup has order 3, the 5-Sylow subgroup has order 5, and any p-Sylow subgroup with  is the trivial subgroup.

Theorem 1. Let $G$ be a finite group. For any prime $p$ , there is a $p$ -Sylow subgroup of $G$ . Also, each $p$ -subgroup of $G$ lies in some $p$ -Sylow subgroup of $G$ .

Theorem 2. For any finite group, for a fixed prime p, all p-Sylow subgroups are conjugate to each other.

Theorem 3. For any finite group G of order $|G|=p^nm$ , where prime p does not divide m, let $n_p$ denote the number of p-Sylow subgroups of G. Then $n_p\equiv 1$ mod $p$ and $n_p|m$ .

Reference- The Sylow Theorems. Keith Conrad.

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