Binary cyclic group

In mathematics, the binary cyclic group of the n-gon is the cyclic group of order 2n, ${\displaystyle C_{2n}}$, thought of as an extension of the cyclic group $C_{n}$ by a cyclic group of order 2. It is the binary polyhedral group corresponding to the cyclic group.^[1]

In terms of binary polyhedral groups, the binary cyclic group is the preimage of the cyclic group of rotations (${\displaystyle C_{n}<\operatorname {SO} (3)}$) under the 2:1 covering homomorphism

$\operatorname {Spin} (3)\to \operatorname {SO} (3)\,$

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 $\operatorname {Spin} (3)\cong \operatorname {Sp} (1)$ where Sp(1) is the multiplicative group of unit quaternions. (For a description of this homomorphism see the article on quaternions and spatial rotations.)

See also

• binary dihedral group
• binary tetrahedral group
• binary octahedral group
• binary icosahedral group

References

1. 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.