Lesson plan
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.)
Article summary
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.)
Key claims
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
Bibliography
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Key claims and bibliography are generated from the same tracked claims and sources as the article page.