Return to Theorems, Glossary, Homework Problems.

**Statement:**

A subgroup of a cyclic group is cyclic.

**Proof:**

Let $G$ be a cyclic group generated by $a$ and let $H$ be a subgroup of $G$. If $H=\{e\}$, then $H=\langle e \rangle$ is cyclic. If $H \neq \{e\}$, then $a^{n} \in H$ for some $n \in \mathbb{Z^{+}}$. Let $m$ be the smallest integer in $\mathbb{Z^{+}}$ such that $a^{m} \in H$.

We claim that $c=a^{m}$ generates $H$; that is,

(1)We must show that every $b \in H$ is a power of $c$. Since $b \in H$ and $H \leq G$, we have $b=a^{n}$ for some $n$. Find $q$ and $r$ such that

(2)in accord with the division algorithm. Then

(3)so

(4)Now since $a^{n} \in H$, $a^{m} \in H$, and $H$ is a group, both $(a^{m})^{-q}$ and $a^{n}$ are in $H$. Thus

(5)that is,

(6)Since $m$ was the smallest positive integer such that $a^{m} \in H$ and $0 \leq r < m$, we must have $r=0$. Thus $n=qm$ and

(7)so $b$ is a power of $c$.