Binary cyclic group explained

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.^[1]

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

2 n ^th roots of unity—that is, the set \left\{\omega ^{k} _{n} | k \in \{ 0 , 1 , 2 ,..., 2 n -1 \}\right\} , where\omega _{n} = e ^{i\pi/n} = \cos (\pi)/(n) + i\sin (\pi)/(n) , using multiplication as the group operation.

See also

• 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

Notes and References

1. .