---
tag: 031M8HX7DA
kind: statement
role: proposition
name: Order of a cyclic subgroup equals the order of its generator
msc: [20A05]
uses: [0YT586TKXF, 0JPYSD5E0B, 0SEX9F9J5Z, 0F5QC101Q7]
sources:
  - url: https://en.wikipedia.org/wiki/Cyclic_group
    locator: Definition and notation
  - url: https://en.wikipedia.org/wiki/Order_(group_theory)
    locator: Order and structure
---

## Statement

Let $G$ be a [[group|G5S1M9]] and let $g \in G$.
If $g$ has finite [[order|0WE2NHZJ7B]] $n$, then $\langle g \rangle = \{e, g, g^2, \ldots, g^{n-1}\}$, these $n$ elements are distinct, and thus $|\langle g \rangle| = n$.
If $g$ has infinite order, then the powers $g^k$ for $k \in \mathbb{Z}$ are distinct, and $\langle g \rangle$ is infinite.

## Remarks

Here $\langle g \rangle$ is the [[cyclic subgroup generated by $g$|08ABT85339]], and $|\langle g \rangle|$ is its [[order|0B85S2V2DR]].
The proof uses [[the divisibility criterion for powers|0F5QC101Q7]]: $g^a = g^b$ exactly when $n$ divides $a - b$.
With [[Lagrange's theorem|3C53TD]], this fact shows that the order of an element of a finite group divides the order of the group.

## Explanations

### L1 human

The two meanings of the word "order" agree.
The order of an element is the number of distinct powers that it has, and these powers form the subgroup that it generates.
For example, a rotation by $120$ degrees has order $3$, and its powers are the three rotations by $0$, $120$ and $240$ degrees.