---
tag: 0YP7WYXF9V
kind: definition
name: Direct product of groups
aliases: [external-direct-product]
msc: ["20E34"]
defines: 00HF4RJ5QN
uses: [G5S1M9]
sources:
  - url: https://en.wikipedia.org/wiki/Direct_product_of_groups
    locator: Definition
---

## Definition

Let $G$ and $H$ be [[groups|G5S1M9]].
The direct product $G \times H$ is the set of ordered pairs $(g, h)$ with $g \in G$ and $h \in H$, with the operation $(g_1, h_1)(g_2, h_2) = (g_1 g_2, h_1 h_2)$.

## Remarks

The identity is $(e_G, e_H)$, and the inverse of $(g, h)$ is $(g^{-1}, h^{-1})$.
If $G$ and $H$ are finite, then $|G \times H| = |G| \cdot |H|$.
The [[order|0WE2NHZJ7B]] of $(g, h)$ is the least common multiple of the orders of $g$ and $h$, when both are finite.
For example, $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z}$ is cyclic of order $6$, but $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$ is not cyclic.

## Explanations

### L1 human

The direct product runs two groups side by side.
An element is a pair, and you multiply the first parts in $G$ and the second parts in $H$.
The two parts never interact.