Binary cyclic group

In mathematics, the binary cyclic group of the n-gon is the cyclic group of order 2n, \( C_{2n} \), 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 (\( C_n < \operatorname{SO}(3)) \) 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

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.

Mathematics Encyclopedia