Binary cyclic group

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the binary cyclic group of the n-gon is the cyclic group of order 2n, 
 
 
 
 
 C
 
 2
 n
 
 
 
 
 {\displaystyle C_{2n}}
 
, thought of as an <a href="/facts/Group_extension/pGoe0l4d">extension</a> of the cyclic group 
 
 
 
 
 C
 
 n
 
 
 
 
 {\displaystyle C_{n}}
 
 by a <a href="/facts/Cyclic_group/JL5zfFuL">cyclic group</a> 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 <a href="/facts/Binary_polyhedral_group/Uc4MRIyl">binary polyhedral group</a> 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
 
 
 <
 SO
 ⁡
 (
 3
 )
 
 
 {\displaystyle C_{n}<\operatorname {SO} (3)}
 
) under the 2:1 <a href="/facts/Covering_homomorphism/IrRjtTL2">covering homomorphism</a>

Spin
        ⁡
        (
        3
        )
        →
        SO
        ⁡
        (
        3
        )
        
      
    
    {\displaystyle \operatorname {Spin} (3)\to \operatorname {SO} (3)\,}

of the <a href="/facts/Special_orthogonal_group/jKWM1KDM">special orthogonal group</a> by the <a href="/facts/Spin_group/tfvR9cqM">spin group</a>.
As a subgroup of the spin group, the binary cyclic group can be described concretely as a discrete subgroup of the unit <a href="/facts/Quaternion/T3tnu7mX">quaternions</a>, under the isomorphism 
 
 
 
 Spin
 ⁡
 (
 3
 )
 ≅
 Sp
 ⁡
 (
 1
 )
 
 
 {\displaystyle \operatorname {Spin} (3)\cong \operatorname {Sp} (1)}
 
 where <a href="/facts/Sp(1)/G4CKBc7z">Sp(1)</a> is the multiplicative group of unit quaternions. (For a description of this homomorphism see the article on <a href="/facts/Quaternions_and_spatial_rotation/8rEnf5sG">quaternions and spatial rotations</a>.)

Binary cyclic group open-in-new

Binary cyclic group