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 $gPg^{-1}=Q$ 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 $gQg^{-1}\subseteq P$.
• Congruence: Let $Syl_{p}(G)$ denote the set of $p$-Sylow subgroups of $G$ and $n_{p}$ denote the cardinality of $Syl_{p}(G)$. Then, $n_{p}\equiv 1\mod p$.