Sylow's theorem

This article gives the statement, and possibly proof, of a basic fact in group theory.
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Statement

Verbal statement

The Sylow's theorem(s) give(s) information about the existence of $p$ -Sylow subgroups of a finite group, as well as the relation among them. More specifically, given a finite group:

Existence: For any prime $p$ , there exists a $p$ -Sylow subgroup
Conjugacy: Any two $p$ -Sylow subgroups are conjugate in the whole group
Domination: Any $p$ -subgroup is contained inside some $p$ -Sylow subgroup
Congruence: The number of $p$ -Sylow subgroups divides the index of any $p$ -Sylow subgroup and is also congruent to $1$ modulo $p$ .

Symbolic statement

Let $G$ be a finite group and $p$ a prime. A subgroup of $G$ is termed a $p$ -Sylow subgroup if its order is a power of $p$ and its index is relatively prime to $p$ . Then Sylow's theorem states that:

Existence: There exists a $p$ -Sylow subgroup $P$ of $G$
Conjugacy: If $P$ and $Q$ are $p$ -Sylow subgroups of $G$ then there exists $g$ in $G$ such that $g P g^{- 1} = Q$ viz $P$ and $Q$ are conjugate subgroups)
Domination: Let $P$ be a $p$ -Sylow subgroup and $Q$ a $p$ -group. Then there exists a $g$ in $G$ such that $g Q g^{- 1} \subseteq P$ .
Congruence: Let $S y l_{p} (G)$ denote the set of $p$ -Sylow subgroups of $G$ and $n_{p}$ denote the cardinality of $S y l_{p} (G)$ . Then, $n_{p} \equiv 1 mod p$ .