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 by a cyclic group of order 2.
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 under the 2:1 covering homomorphism
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 where Sp(1) is the multiplicative group of unit quaternions. (For a description of this homomorphism see the article on quaternions and spatial rotations.)
Famous quotes containing the word group:
“Belonging to a group can provide the child with a variety of resources that an individual friendship often cannot—a sense of collective participation, experience with organizational roles, and group support in the enterprise of growing up. Groups also pose for the child some of the most acute problems of social life—of inclusion and exclusion, conformity and independence.”
—Zick Rubin (20th century)