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 of order 4 $\mathbb{Z}/4$, the Klein group $\mathbb{Z}/2 \times \mathbb{Z}/2$ is the only other group of order 4, up to isomorphism. (This follows, for instance, by the fundamental theorem of finitely generated abelian groups, as in this example).

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**

See also

- Wikipedia,
*Klein four-group*

Last revised on October 24, 2020 at 09:20:13. See the history of this page for a list of all contributions to it.