Binary cyclic group

In mathematics, the binary cyclic group of the n-gon is the cyclic group of order 2n, ${\displaystyle C_{2n}}$, thought of as an extension of the cyclic group ${\displaystyle C_{n}}$ by a cyclic group of order 2. Coxeter writes the binary cyclic group with angle-brackets, ⟨n⟩, and the index 2 subgroup as (n) or [n]⁺.

It is the binary polyhedral group corresponding to the cyclic group.^[1]

In terms of binary polyhedral groups, the binary cyclic group is the preimage of the cyclic group of rotations (${\displaystyle C_{n}<\operatorname {SO} (3)}$) under the 2:1 covering homomorphism

${\displaystyle \operatorname {Spin} (3)\to \operatorname {SO} (3)\,}$

of the special orthogonal group by the spin group.

As a subgroup of the spin group, the binary cyclic group can be described concretely as a discrete subgroup of the unit quaternions, under the isomorphism ${\displaystyle \operatorname {Spin} (3)\cong \operatorname {Sp} (1)}$ where Sp(1) is the multiplicative group of unit quaternions. (For a description of this homomorphism see the article on quaternions and spatial rotations.)

The binary cyclic group can be defined as the set of ${\displaystyle 2n}$^th roots of unity—that is, the set ${\displaystyle \left\{\omega _{n}^{k}\;|\;k\in \{0,1,2,...,2n-1\}\right\}}$, where

${\displaystyle \omega _{n}=e^{i\pi /n}=\cos {\frac {\pi }{n}}+i\sin {\frac {\pi }{n}},}$

using multiplication as the group operation.

• binary dihedral group, ⟨2,2,n⟩, order 4n
• binary tetrahedral group, ⟨2,3,3⟩, order 24
• binary octahedral group, ⟨2,3,4⟩, order 48
• binary icosahedral group, ⟨2,3,5⟩, order 120