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

**Classical groups**

**Finite groups**

**Group schemes**

**Topological groups**

**Lie groups**

**Super-Lie groups**

**Higher groups**

**Cohomology and Extensions**

**Related concepts**

One calls a cyclic group of order an even number $2(n+1)$ ($n \in \mathbb{N}$) a *binary cyclic group* if one thinks of it as being the central extension of a cyclic group by the group of order 2:

$\array{
\mathbb{Z}/2\mathbb{Z} &\hookrightarrow& \mathbb{Z}/2(n+1)\mathbb{Z}
\\
&& \downarrow
\\
&& \mathbb{Z}/(n+1)\mathbb{Z}
}$

This way binary cyclic groups are related to cyclic groups as the binary dihedral groups are related to the dihedral groups. Note that if $n+1$ is odd, then the extension is (necessarily) the trivial extension, and if $n+1$ is even, then this extension is the (unique) nontrivial extension.

From the point of view of the classification of finite rotation groups, the cyclic groups, dihedral groups etc. are the finite subgroups of the special orthogonal group $SO(3)$, while their binary versions are those subgroups of the special unitary group $SU(2)$ that arise from the former by taking preimages under the double cover-projection $SU(2) \simeq Spin(3) \to SO(3)$.

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

See also

- Wikipedia,
*Binary cyclic group*

Last revised on April 17, 2018 at 05:47:57. See the history of this page for a list of all contributions to it.