# Sylow's theorem

## Contents

This article gives the statement, and possibly proof, of a basic fact in group theory.
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This fact is related to: Sylow 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 $p$.

### Statement with symbols

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 $p$. Note that by Lagrange's theorem, the order of a Sylow subgroup is the largest power of $p$ dividing the order of $G$, and thus, it is a multiple of the order of any $p$-subgroup (subgroup whose order is a power of $p$) of $G$.

Then Sylow's theorem states that:

1. Existence: There exists a $p$-Sylow subgroup $P$ of $G$
2. Conjugacy: If $P$ and $Q$ are $p$-Sylow subgroups of $G$ then there exists $g$ in $G$ such that $gPg^{-1} = Q$ (i.e., $P$ and $Q$ are conjugate subgroups)
3. Domination: Let $P$ be a $p$-Sylow subgroup and $Q$ a $p$-group. Then there exists a $g$ in $G$ such that $gQg^{-1} \subseteq P$.
4. 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, $n_p \equiv 1\mod p$.

## Related facts

### Corollaries

There are other corollaries too, many of which factor through these corollaries.

## Proof breakup

1. Existence: For full proof, refer: Sylow subgroups exist
2. Conjugacy: This follows from Domination, though it can also be proved through the methods used to establish Congruence. For full proof, refer: Sylow implies order-conjugate
3. Domination: For full proof, refer: Sylow implies order-dominating
4. Congruence: For full proof, refer: Congruence condition on Sylow numbers