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.)
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“It is not God that is worshipped but the group or authority that claims to speak in His name. Sin becomes disobedience to authority not violation of integrity.”
—Sarvepalli, Sir Radhakrishnan (1888–1975)