---
tag: 09GCQ5STVJ
kind: statement
role: theorem
name: Second Sylow theorem
msc: [20D20]
uses: [0JWZX6FKSH, 0D6WHTTH7T, 0V98WY94B8]
sources:
  - url: https://en.wikipedia.org/wiki/Sylow_theorems
    locator: Statement
---

## Statement

Let $G$ be a finite [[group|G5S1M9]] and let $p$ be a prime number.
Then any two [[Sylow p-subgroups|0053CQE9Y2]] $P$ and $Q$ of $G$ are [[conjugate|0H11ZPX4ND]]: there is a $g \in G$ with $Q = gPg^{-1}$.
Moreover, each $p$-subgroup of $G$ is contained in some Sylow $p$-subgroup of $G$.

## Remarks

So all Sylow $p$-subgroups of $G$ are [[isomorphic|08MG7H8JW5]].
A Sylow $p$-subgroup is [[normal|0CB0DX5BVV]] in $G$ if and only if $n_p = 1$.
The proof lets a $p$-subgroup $H$ act on the left cosets of $P$ and uses the [[fixed point lemma|098D2E7DEH]].

## Explanations

### L1 human

The Sylow $p$-subgroups of a group can look different as subsets.
But each one can be moved onto each other one by a conjugation.
So, up to a renaming, there is only one Sylow $p$-subgroup.