# Sylow’s Theorem (Summary)

In this post we review Sylow’s theorem and as an example we solve the following problem.

## Problem 64

Show that a group of order $200$ has a normal Sylow $5$-subgroup.

## Review of Sylow’s Theorem

One of the important theorems in group theory is Sylow’s theorem.

Sylow’s theorem is a very powerful tool to solve the classification problem of finite groups of a given order.

In this article, we review several terminologies, the contents of Sylow’s theorem, and its corollary.
We also give an example that can be solved using Sylow’s theorem.

At the end of this post, the links to various Sylow’s theorem problems are given.

We first introduce several definitions.

### Definition 1.

Let $G$ be a group and $p$ be a prime number.

1. A group of order $p^{\alpha}$ for some non-negative integer $\alpha$ is called a $p$-group.
2. A subgroup of $G$ which is a $p$-subgroup is called $p$-subgroup.

### Definition 2.

Let $G$ be a finite group of order $n$. Let $p$ be a prime number dividing $n$.
Write $n=p^{\alpha}m$, where $\alpha, m \in \Z$ and $p$ does not divide $m$.
Then any subgroup $H$ of $G$ is called a Sylow $p$-group of $G$ if the order of $H$ is $p^{\alpha}$.

## Sylow’s theorem

Let $G$ be a finite group of order $p^{\alpha}m$, where the prime number $p$ does not divide $m$.

1. There exists at least one Sylow $p$-subgroup of $G$.
2. If $P$ is a Sylow $p$-subgroup of $G$ and $Q$ is any $p$-subgroup of $G$, then there exists $g \in G$ such that $Q$ is a subgroup of $gPg^{-1}$.
In particular, any two Sylow $p$-subgroups of $G$ are conjugate in $G$.
3. The number $n_p$ of Sylow $p$-subgroups of $G$ is
$n_p \equiv 1 \pmod p.$ That is, $n_p=pk+1$ for some $k\in \Z$.
4. The number $n_p$ of Sylow $p$-subgroup of $G$ is the index of the normalizer $N_G(P)$ in $G$ for any Sylow $p$-subgroup $P$, hence $n_p$ divides $m$.

### Corollary

In the notation of the previous theorem, if the number $n_p$ of Sylow $p$-subgroup of $G$ is $n_p=1$, then the Sylow $p$-subgroup is a normal subgroup of $G$.

## Example/Problem.

Now as an example we solve the problem.

### Problem.

Show that a group of order $200$ has a normal Sylow $5$-subgroup.

### Solution.

We have the factorization $200=2^3\cdot 5^2$.

By Sylow’s theorem the number of Sylow $5$-subgroup satisfies $n_5 \equiv 1 \pmod 5$ and $n_5$ divides $8$.

The numbers satisfies $n_5 \equiv 1 \pmod 5$ are $n_5=1, 6, 11, \cdots$.
Among these numbers, only $1$ divides $8$.

Thus the only number satisfies both conditions is $1$.
Hence $n_5=1$ and there is only one Sylow $5$-subgroup.

Then by the corollary, the Sylow $5$-subgroup is normal.

## More Problems on Sylow’s theorem

Sylow’s theorem is a handy tool to determine the group structure of a finite group.
We list here several problems/examples that can be solved using Sylow’s theorem.

All solutions are given in the links below.

### More from my site

• Group of Order $pq$ Has a Normal Sylow Subgroup and Solvable Let $p, q$ be prime numbers such that $p>q$. If a group $G$ has order $pq$, then show the followings. (a) The group $G$ has a normal Sylow $p$-subgroup. (b) The group $G$ is solvable.   Definition/Hint For (a), apply Sylow's theorem. To review Sylow's theorem, […]
• Are Groups of Order 100, 200 Simple? Determine whether a group $G$ of the following order is simple or not. (a) $|G|=100$. (b) $|G|=200$.   Hint. Use Sylow's theorem and determine the number of $5$-Sylow subgroup of the group $G$. Check out the post Sylow’s Theorem (summary) for a review of Sylow's […]
• Sylow Subgroups of a Group of Order 33 is Normal Subgroups Prove that any $p$-Sylow subgroup of a group $G$ of order $33$ is a normal subgroup of $G$.   Hint. We use Sylow's theorem. Review the basic terminologies and Sylow's theorem. Recall that if there is only one $p$-Sylow subgroup $P$ of $G$ for a fixed prime $p$, then $P$ […]
• Group of Order 18 is Solvable Let $G$ be a finite group of order $18$. Show that the group $G$ is solvable.   Definition Recall that a group $G$ is said to be solvable if $G$ has a subnormal series $\{e\}=G_0 \triangleleft G_1 \triangleleft G_2 \triangleleft \cdots \triangleleft G_n=G$ such […]
• A Group of Order $pqr$ Contains a Normal Subgroup of Order Either $p, q$, or $r$ Let $G$ be a group of order $|G|=pqr$, where $p,q,r$ are prime numbers such that $p<q<r$. Show that $G$ has a normal subgroup of order either $p,q$ or $r$. Hint. Show that using Sylow's theorem that $G$ has a normal Sylow subgroup of order either $p,q$, or $r$. Review […]
• Prove that a Group of Order 217 is Cyclic and Find the Number of Generators Let $G$ be a finite group of order $217$. (a) Prove that $G$ is a cyclic group. (b) Determine the number of generators of the group $G$.     Sylow's Theorem We will use Sylow's theorem to prove part (a). For a review of Sylow's theorem, check out the […]
• Subgroup Containing All $p$-Sylow Subgroups of a Group Suppose that $G$ is a finite group of order $p^an$, where $p$ is a prime number and $p$ does not divide $n$. Let $N$ be a normal subgroup of $G$ such that the index $|G: N|$ is relatively prime to $p$. Then show that $N$ contains all $p$-Sylow subgroups of […]
• Non-Abelian Group of Order $pq$ and its Sylow Subgroups Let $G$ be a non-abelian group of order $pq$, where $p, q$ are prime numbers satisfying $q \equiv 1 \pmod p$. Prove that a $q$-Sylow subgroup of $G$ is normal and the number of $p$-Sylow subgroups are $q$.   Hint. Use Sylow's theorem. To review Sylow's theorem, check […]

### 18 Responses

1. 08/18/2016

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2. 08/21/2016

[…] Review Sylow’s theorem (Especially (3) and (4) in the theorem). […]

3. 12/19/2016

[…] For more details, check out the post Sylow’s theorem (summary) […]

4. 12/21/2016

[…] To review Sylow’s theorem, read the post Sylow’s Theorem (summary) […]

5. 01/06/2017

[…] For (a), apply Sylow’s theorem. To review Sylow’s theorem, read the post Sylow’s Theorem (summary). […]

6. 01/06/2017

[…] $18=2cdot 3^2$, the number $n_3$ of Sylow $3$-subgroups is $1$ by the Sylow theorem. (Sylow’s theorem implies that $n_3 equi 1 pmod{3}$ and $n_3$ divides $2$.) Hence the […]

7. 01/07/2017

[…] Sylow’s theorem and determine the number of $5$-Sylow subgroup of the group $G$. Check out Sylow’s Theorem (summary) for a review of Sylow’s […]

8. 01/29/2017

[…] We use Sylow’s theorem. Review the basic terminologies and Sylow’s theorem. […]

9. 02/06/2017

[…] the post summary of Sylow’s Theorem to review Sylow’s […]

10. 02/12/2017

[…] Sylow’s theorem. To review Sylow’s theorem, check out the post Sylow’s Theorem (summary). Read the corollary there as well to understand the proof […]

11. 06/15/2017

[…] For a review of Sylow’s theorem, check out the post “Sylow’s Theorem (summary)“. […]

12. 06/18/2017

[…] that $20449=11^2 cdot 13^2$. Let $G$ be a group of order $20449$. We prove by Sylow’s theorem that there are a unique Sylow $11$-subgroup and a unique Sylow $13$-subgroup of $G$. Hence $G$ is […]

13. 06/20/2017

[…] You may use Sylow’s theorem. For a review of Sylow’s theorem, please check out the post Sylow’s Theorem (summary). […]

14. 06/20/2017

[…] out the post “Sylow’s Theorem (summary)” for a review of Sylow’s […]

15. 06/24/2017

[…] For a review of the Sylow’s theorem, check out the post “Sylow’s Theorem (summary)“. […]

16. 07/06/2017

[…] Use the conjugate part of the Sylow theorem. See the second statement of the Sylow theorem. […]

17. 09/19/2017

[…] Sylow’s theorem. (See Sylow’s Theorem (Summary) for a review of Sylow’s […]

18. 12/14/2017

[…] Sylow’s theorem, we know that [n_{19} equiv 1 pmod{19} text{ and } n_{19} mid 3.] It follows that […]

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##### All the Conjugacy Classes of the Dihedral Group $D_8$ of Order 8

Determine all the conjugacy classes of the dihedral group $D_{8}=\langle r,s \mid r^4=s^2=1, sr=r^{-1}s\rangle$ of order $8$.

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