Klein four-group

- group, ∞-group
- group object, group object in an (∞,1)-category
- abelian group, spectrum
- group action, ∞-action
- representation, ∞-representation
- progroup
- homogeneous space

The direct product group of the group of order 2 with itself is known as the *Klein four group*:

$\mathbb{Z}/2 \times \mathbb{Z}/2
\,.$

Besides the cyclic group $\mathbb{Z}/4$ of order 4, the Klein group $\mathbb{Z}/2 \times \mathbb{Z}/2$ is the only other group of order 4, up to isomorphism.

In particular the Klein group is *not* itself a cyclic group, and it is in fact the smallest non-trivial group which is not a cyclic group.

In the ADE-classification of finite subgroups of SO(3), the Klein four-group is the smallest in the D-series, labeled by D4.

**ADE classification** and **McKay correspondence**

Dynkin diagram/ Dynkin quiver | Platonic solid | finite subgroups of SO(3) | finite subgroups of SU(2) | simple Lie group |
---|---|---|---|---|

$A_{n \geq 1}$ | cyclic group $\mathbb{Z}_{n+1}$ | cyclic group $\mathbb{Z}_{n+1}$ | special unitary group $SU(n+1)$ | |

D4 | Klein four-group $D_4 \simeq \mathbb{Z}_2 \times \mathbb{Z}_2$ | quaternion group $2 D_4 \simeq$ Q8 | SO(8) | |

$D_{n \geq 4}$ | dihedron, hosohedron | dihedral group $D_{2(n-2)}$ | binary dihedral group $2 D_{2(n-2)}$ | special orthogonal group $SO(2n)$ |

$E_6$ | tetrahedron | tetrahedral group $T$ | binary tetrahedral group $2T$ | E6 |

$E_7$ | cube, octahedron | octahedral group $O$ | binary octahedral group $2O$ | E7 |

$E_8$ | dodecahedron, icosahedron | icosahedral group $I$ | binary icosahedral group $2I$ | E8 |

See also

- Wikipedia,
*Klein four-group*

Last revised on February 18, 2019 at 08:03:30. See the history of this page for a list of all contributions to it.