Binary cyclic group explained
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. 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.[1]
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\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.)
Presentation
The binary cyclic group can be defined as the set of
th roots of unity—that is, the set
| k\in\{0,1,2,...,2n-1\}\right\}
, where
\omegan=ei\pi/n=\cos
+i\sin
,
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
Notes and References
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