---
tag: 0PMRQJVTME
kind: object
name: Klein four-group
aliases: [vierergruppe]
msc: [20K01]
uses: [071DDFVG4J, 0YP7WYXF9V]
sources:
  - url: https://en.wikipedia.org/wiki/Klein_four-group
    locator: lead section
  - url: https://en.wikipedia.org/wiki/Klein_four-group
    locator: Permutation representation
same_as:
  wikidata: Q550593
  wikipedia: Klein four-group
---

## Description

The Klein four-group $V$ is the group with four elements $e, a, b, c$, in which $e$ is the identity and each element is its own inverse.
The product of two distinct elements of $\{a, b, c\}$ is the third one.
It is [[abelian|082GHNPJ53]], and it is [[isomorphic|08MG7H8JW5]] to the [[direct product|00HF4RJ5QN]] $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$.

## Remarks

The Klein four-group is the smallest group that is not [[cyclic|0KVGH0J5B3]], because it has no element of order $4$.
It is the symmetry group of a rectangle that is not a square.
In $S_4$, the set $\{e, (1\ 2)(3\ 4), (1\ 3)(2\ 4), (1\ 4)(2\ 3)\}$ is a normal subgroup isomorphic to $V$.
Each group of order $4$ is isomorphic to $\mathbb{Z}/4\mathbb{Z}$ or to $V$.

## Explanations

### L1 human

The Klein four-group is like two light switches that you can flip separately.
Each switch flipped two times is back to the start.
Flipping both switches is the third element that is not the identity.