Binary cyclic group
In mathematics, the binary cyclic group of the n-gon is the cyclic group of order 2n,, thought of as an extension of the cyclic group by a cyclic group of order 2. Coxeter writes the binary cyclic group with angle-brackets, ⟨n⟩, and the index 2 subgroup as or +.
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 under the 2:1 covering homomorphism
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 where Sp is the multiplicative group of unit quaternions.Presentation
The binary cyclic group can be defined as:
