group of order 2

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

There is, up to isomorphism, a unique simple group of order 2:

it has two elements $(1,\sigma)$, where $\sigma \cdot \sigma = 1$.

This is usually denoted $\mathbb{Z}_2$ or $\mathbb{Z}/2\mathbb{Z}$, because it is the cokernel (the quotient by the image of) the homomorphism

$\cdot 2 : \mathbb{Z} \to \mathbb{Z}$

on the additive group of integers. As such $\mathbb{Z}_2$ is the special case of a cyclic group $\mathbb{Z}_p$ for $p = 2$ and hence also often denoted $C_2$.

In the ADE-classification of finite subgroups of SU(2), the group of order 2 is the smallest non-trivial group, and the smallest in the A-series:

**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 |

Last revised on May 8, 2019 at 10:53:02. See the history of this page for a list of all contributions to it.