Note the prime factorization $217=7\cdot 31$.
We first determine the number $n_p$ of Sylow $p$-group for $p=7, 31$.
Recall from Sylow’s theorem that
\begin{align*}
&n_p \equiv 1 \pmod{p}\\[6pt]
&n_p \text{ divides } n/p.
\end{align*}

Thus, $n_7$ could be $1, 8, 15, 22, 29,\dots$ and $n_7$ needs to divide $217/7=31$.
Hence the only possible value for $n_7$ is $n_7=1$.
So there is a unique Sylow $7$-subgroup $P_7$ of $G$.

By Sylow’s theorem, the unique Sylow $7$-subgroup must be a normal subgroup of $G$.

Similarly, $n_{31}=1, 32, \dots$ and $n_{31}$ must divide $217/31=7$, and hence we must have $n_{31}=1$.
Thus $G$ has a unique normal Sylow $31$-subgroup $P_{31}$.

Note that these Sylow subgroup have prime order, and hence they are isomorphic to cyclic groups:
\[P_7\cong \Zmod{7} \text{ and } P_{31}\cong \Zmod{31}.\]

It is also straightforward to see that $P_7 \cap P_{31}=\{e\}$, where $e$ is the identity element in $G$.

In summary, we have

$P_7, P_{31}$ are normal subgroups of $G$.

$P_7 \cap P_{31}=\{e\}$.

$|P_7P_{31}|=|G|$.

These yields that $G$ is a direct product of $P_7$ and $P_{31}$, and we obtain
\[G=P_7\times P_{31}\cong \Zmod{7} \times \Zmod{31}\cong \Zmod{217}.\]
Hence $G$ is a cyclic group.

(b) Determine the number of generators of the group $G$.

Recall that the number of generators of a cyclic group of order $n$ is equal to the number of integers between $1$ and $n$ that are relatively prime to $n$.
Namely, the number of generators is equal to $\phi(n)$, where $\phi$ is the Euler totient function.

By part (a), we know that $G$ is a cyclic group of order $217$.
Thus, the number of generators of $G$ is
\begin{align*}
\phi(217)=\phi(7)\phi(31)=6\cdot 30=180,
\end{align*}
where the first equality follows since $\phi$ is multiplicative.

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 […]

A Simple Abelian Group if and only if the Order is a Prime Number
Let $G$ be a group. (Do not assume that $G$ is a finite group.)
Prove that $G$ is a simple abelian group if and only if the order of $G$ is a prime number.
Definition.
A group $G$ is called simple if $G$ is a nontrivial group and the only normal subgroups of $G$ is […]

Every Group of Order 20449 is an Abelian Group
Prove that every group of order $20449$ is an abelian group.
Outline of the Proof
Note 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 […]

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 […]

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 […]

Fundamental 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 […]

Surjective Group Homomorphism to $\Z$ and Direct Product of Abelian Groups
Let $G$ be an abelian group and let $f: G\to \Z$ be a surjective group homomorphism.
Prove that we have an isomorphism of groups:
\[G \cong \ker(f)\times \Z.\]
Proof.
Since $f:G\to \Z$ is surjective, there exists an element $a\in G$ such […]

## 1 Response

[…] Prove that a Group of Order 217 is Cyclic and Find the Number of Generators […]