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
Coxeter writes the binary cyclic group with angle-brackets, ⟨n⟩, and the index 2 subgroup as (n) or [n]+.
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(1) is the multiplicative group of unit quaternions.
(For a description of this homomorphism see the article on quaternions and spatial rotations.)
The binary cyclic group can be defined as the set of