---
tag: 0KB1ND7XDS
kind: statement
role: theorem
name: Subgroups of cyclic groups are cyclic
msc: [20K27]
uses: [0B85Q41RP6, 0N0PXCTYA7]
sources:
  - url: https://en.wikipedia.org/wiki/Cyclic_group
    locator: Subgroups
---

## Statement

Let $G$ be a [[cyclic group|0KVGH0J5B3]] and let $H$ be a [[subgroup|0X1JXT47GA]] of $G$.
Then $H$ is cyclic.

## Remarks

Let $g$ generate $G$.
If $H \neq \{e\}$, let $m$ be the smallest positive integer with $g^m \in H$; then division with remainder shows that $H = \langle g^m \rangle$.
For example, each subgroup of $(\mathbb{Z}, +)$ has the form $m\mathbb{Z}$, generated by $m$.

## Explanations

### L1 human

A subgroup of a cyclic group is generated by the smallest positive power of the generator that it contains.
Any other power in the subgroup is a multiple of this one, or the subgroup would contain a smaller positive power.
Thus the subgroup is again cyclic.