Binary cyclic group

In mathematics, the binary cyclic group of the n-gon is the cyclic group of order 2n, $$C_{2n}$$, thought of as an extension of the cyclic group $$C_n$$ 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.

In terms of binary polyhedral groups, the binary cyclic group is the preimage of the cyclic group of rotations ($$C_n < \operatorname{SO}(3)$$) 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 $$\left\{\omega_n^k \; | \; k \in \{0,1,2,...,2n-1\}\right\}$$, where
 * $$\omega_n = e^{i\pi/n} = \cos\frac{\pi}{n} + i\sin\frac{\pi}{n},$$

using multiplication as the group operation.