---
tag: 0YT586TKXF
kind: definition
name: Cyclic subgroup generated by an element
aliases: [cyclic-subgroup]
msc: [20A05]
defines: 08ABT85339
uses: [0N0PXCTYA7, 00DRK90Z6K]
sources:
  - url: https://en.wikipedia.org/wiki/Cyclic_group
    locator: Definition and notation
---

## Definition

Let $G$ be a [[group|G5S1M9]] and let $g \in G$.
The cyclic subgroup generated by $g$ is the set $\langle g \rangle = \{g^k : k \in \mathbb{Z}\}$ of all integer powers of $g$.

## Remarks

By [[the laws of exponents|00DRK90Z6K]], $g^j (g^k)^{-1} = g^{j-k}$, thus $\langle g \rangle$ is a [[subgroup|0X1JXT47GA]] of $G$.
It is the [[subgroup generated by|0RP7VWQ0YN]] the set $\{g\}$.
In additive notation, $\langle g \rangle = \{kg : k \in \mathbb{Z}\}$, for example $\langle 3 \rangle = 3\mathbb{Z}$ in $(\mathbb{Z}, +)$.

## Explanations

### L1 human

Take one element and all its powers, positive and negative, and the identity.
These powers form a subgroup, the smallest one that contains the element.
In the integers with addition, the cyclic subgroup generated by $3$ is the set of multiples of $3$.