Binary cyclic group explained

In mathematics, the binary cyclic group of the n-gon is the cyclic group of order 2n,

C2n

, thought of as an extension of the cyclic group

Cn

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 (

Cn<\operatorname{SO}(3)

) under the 2:1 covering homomorphism

\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

\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.)

Presentation

The binary cyclic group can be defined as the set of

2n

th roots of unity—that is, the set
k
\left\{\omega
n

|k\in\{0,1,2,...,2n-1\}\right\}

, where

\omegan=ei\pi/n=\cos

\pi
n

+i\sin

\pi
n

,

using multiplication as the group operation.

See also

Notes and References

  1. .