# Group of Order $pq$ Has a Normal Sylow Subgroup and Solvable

## Problem 245

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, read the post Sylow’s Theorem (summary).

In particular, we will use Sylow’s theorem (3) and (4), and its corollary in the proof below.

For (b), recall that a group $G$ is solvable if $G$ has a subnormal series
$\{e\}=G_0 \triangleleft G_1 \triangleleft G_2 \triangleleft \cdots \triangleleft G_n=G$ such that the factor groups $G_i/G_{i-1}$ are all abelian groups for $i=1,2,\dots, n$.

## Proof.

### (a) The group $G$ has a normal Sylow $p$-subgroup

By Sylow’s theorem, the number $n_p$ of Sylow $p$-subgroups of $G$ satisfies $n_p\equiv 1 \pmod{p}$ and $n_p$ divides $q$.
The only such number is $n_p=1$.

Thus $G$ has the unique Sylow $p$-subgroup $P$ of order $p$.
Since $P$ is the unique Sylow $p$-subgroup, it is a normal subgroup of $G$.

### (b) The group $G$ is solvable

Let $P$ be the normal Sylow subgroup of $G$ obtained in (a).
Then we have the following subnormal series
$\{e\} \triangleleft P \triangleleft G,$ where $e$ is the identity element of $G$.

The factor groups are $G/P$ and $P/\{e\}\cong P$.
The order of the group $P$ is the prime $p$, and hence $P$ is an abelian group.
The order $|G/P|=|G|/|P|=pq/q=q$ is also a prime, and thus $G/P$ is an abelian group.
Thus the factor groups are abelian. Thus $G$ is a solvable group.

## Related Question.

The similar problems are

### More from my site

• A Group of Order $20$ is Solvable Prove that a group of order $20$ is solvable.   Hint. Show that a group of order $20$ has a unique normal $5$-Sylow subgroup by Sylow's theorem. See the post summary of Sylow’s Theorem to review Sylow's theorem. Proof. Let $G$ be a group of order $20$. The […]
• 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 […]
• 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 […]
• 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 $P$ […]
• If a Sylow Subgroup is Normal in a Normal Subgroup, it is a Normal Subgroup Let $G$ be a finite group. Suppose that $p$ is a prime number that divides the order of $G$. Let $N$ be a normal subgroup of $G$ and let $P$ be a $p$-Sylow subgroup of $G$. Show that if $P$ is normal in $N$, then $P$ is a normal subgroup of $G$.   Hint. It follows from […]
• Every Group of Order 12 Has a Normal Subgroup of Order 3 or 4 Let $G$ be a group of order $12$. Prove that $G$ has a normal subgroup of order $3$ or $4$.   Hint. Use Sylow's theorem. (See Sylow’s Theorem (Summary) for a review of Sylow's theorem.) Recall that if there is a unique Sylow $p$-subgroup in a group $GH$, then it is […]
• 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 […]
• 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 […]

### 2 Responses

1. 01/06/2017

[…] Group of order pq has a normal Sylow subgroup and solvable […]

2. 01/06/2017

[…] Group of order pq has a normal Sylow subgroup and solvable […]

##### Pullback Group of Two Group Homomorphisms into a Group

Let $G_1, G_1$, and $H$ be groups. Let $f_1: G_1 \to H$ and $f_2: G_2 \to H$ be group homomorphisms....

Close