{{Short description|Algebraic structure}} In mathematics, the '''binary cyclic group''' of the ''n''-gon is the cyclic group of order 2''n'', <math>C_{2n}</math>, thought of as an extension of the cyclic group <math>C_n</math> 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>+</sup>.
It is the binary polyhedral group corresponding to the cyclic group.<ref>{{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}}.</ref>
In terms of binary polyhedral groups, the binary cyclic group is the preimage of the cyclic group of rotations (<math>C_n < \operatorname{SO}(3)</math>) under the 2:1 covering homomorphism :<math>\operatorname{Spin}(3) \to \operatorname{SO}(3)\,</math> 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 <math>\operatorname{Spin}(3) \cong \operatorname{Sp}(1)</math> 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 <math>2n</math><sup>th</sup> roots of unity—that is, the set <math>\left\{\omega_n^k \; | \; k \in \{0,1,2,...,2n-1\}\right\}</math>, where :<math>\omega_n = e^{i\pi/n} = \cos\frac{\pi}{n} + i\sin\frac{\pi}{n},</math> 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== {{reflist}}
Cyclic