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.{{citation

| last = Coxeter | first = H. S. M. | authorlink = Harold Scott MacDonald Coxeter

| contribution = Symmetrical definitions for the binary polyhedral groups

| mr = 0116055

| pages = 64–87

| publisher = American Mathematical Society

| location = Providence, R.I.

| title = Proc. Sympos. Pure Math., Vol. 1

| url = https://books.google.com/books?id=V_iX4A5RBtoC&pg=PA64

| year = 1959}}.

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.

References

Cyclic

binary cyclic group

Presentation

See also

References