In mathematics, the binary cyclic group of the n-gon is the cyclic group of order 2n, C 2 n {\displaystyle C_{2n}} , thought of as an extension of the cyclic group C n {\displaystyle 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.
In terms of binary polyhedral groups, the binary cyclic group is the preimage of the cyclic group of rotations ( C n < SO ( 3 ) {\displaystyle C_{n}<\operatorname {SO} (3)} ) under the 2:1 covering homomorphism
Spin ( 3 ) → SO ( 3 ) {\displaystyle \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 Spin ( 3 ) ≅ Sp ( 1 ) {\displaystyle \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 {\displaystyle 2n} th roots of unity—that is, the set { ω n k | k ∈ { 0 , 1 , 2 , . . . , 2 n − 1 } } {\displaystyle \left\{\omega _{n}^{k}\;|\;k\in \{0,1,2,...,2n-1\}\right\}} , where
ω n = e i π / n = cos π n + i sin π n , {\displaystyle \omega _{n}=e^{i\pi /n}=\cos {\frac {\pi }{n}}+i\sin {\frac {\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
References
Coxeter, H. S. M. (1959), "Symmetrical definitions for the binary polyhedral groups", Proc. Sympos. Pure Math., Vol. 1, Providence, R.I.: American Mathematical Society, pp. 64–87, MR 0116055. /wiki/Harold_Scott_MacDonald_Coxeter ↩