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. 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 ) 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.)

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. 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 ) 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.) \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==

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