Binary Cyclic Group

In mathematics, the binary cyclic group of the ''n''-gon is the cyclic group of order 2''n'', $C_$, 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''sup>+.

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 ($C_n < \operatorname(3)$) under the 2:1 covering homomorphism
:$\operatorname(3) \to \operatorname(3)\,$
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 $\operatorname(3) \cong \operatorname(1)$ 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 $\left\$, where
:$\omega_n = e^ = \cos\frac + i\sin\frac,$
using multiplication as the group operation.

See also

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

References

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Cyclic