Group of Order 18 is Solvable

Group Theory Problems and Solutions in Mathematics

Problem 118

Let $G$ be a finite group of order $18$.

Show that the group $G$ is solvable.
 
LoadingAdd to solve later

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 that the factor groups $G_i/G_{i-1}$ are all abelian groups for $i=1,2,\dots, n$.

Proof.

Since $18=2\cdot 3^2$, the number $n_3$ of Sylow $3$-subgroups is $1$ by the Sylow theorem.
(Sylow’s theorem implies that $n_3 \equiv 1 \pmod{3}$ and $n_3$ divides $2$.)
Hence the unique Sylow $3$-subgroup $P$ is a normal subgroup of $G$.

The order of $P$ is $9$, a square of a prime number, thus $P$ is abelian.
(See A group of order the square of a prime is abelian.)
Also, the order of the quotient group $G/P$ is $2$, thus $G/P$ is an abelian (cyclic) group.

Thus we have a filtration
\[G \triangleright P \triangleright \{e\}\] whose factors $G/P, P/\{e\}$ are abelian groups, hence $G$ is solvable.

Related Question.

Check the following similar questions.


LoadingAdd to solve later

More from my site

  • A Group of Order $20$ is SolvableA 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 $pq$ Has a Normal Sylow Subgroup and SolvableGroup 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, […]
  • Fundamental Theorem of Finitely Generated Abelian Groups and its applicationFundamental Theorem of Finitely Generated Abelian Groups and its application In this post, we study the Fundamental Theorem of Finitely Generated Abelian Groups, and as an application we solve the following problem. Problem. Let $G$ be a finite abelian group of order $n$. If $n$ is the product of distinct prime numbers, then prove that $G$ is isomorphic […]
  • Sylow Subgroups of a Group of Order 33 is Normal SubgroupsSylow 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 $P$ […]
  • A Group of Order $pqr$ Contains a Normal Subgroup of Order Either $p, q$, or $r$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 […]
  • Are Groups of Order 100, 200 Simple?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 […]
  • Non-Abelian Group of Order $pq$ and its Sylow SubgroupsNon-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 […]
  • Every Group of Order 12 Has a Normal Subgroup of Order 3 or 4Every 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 […]

You may also like...

1 Response

Leave a Reply

Your email address will not be published. Required fields are marked *

More in Group Theory
Group Theory Problems and Solutions in Mathematics
If a Subgroup Contains a Sylow Subgroup, then the Normalizer is the Subgroup itself

Let $G$ be a finite group and $P$ be a nontrivial Sylow subgroup of $G$. Let $H$ be a subgroup...

Close