Covering of the orthogonal group

[Jason Bennett]

Homework Statement

see title

Relevant Equations

see below

Progress:


𝜙:𝑂(3)→ℤ2


𝜓:𝑂(3)→𝑆𝑂(3)


𝜃:𝑂(3)/𝑆𝑂(3)→ℤ2



𝜙(𝑂)=det(𝑂)

with 𝑂∈𝑂(3), that way

𝜙(𝑂)↦{−1,1}≅ℤ2,

where 1 is the identity element.


Ker(𝜙) = {𝑂∈𝑆𝑂(3)|𝜙(𝑂)=1}=𝑆𝑂(3), since det(𝑂)=1 for 𝑂∈𝑆𝑂(3).


By the multiplicative property of the determinant function, 𝜙 = homomorphism.



***What is the form of the canonical homomorphism (𝜓) in this case?


I'm used to the coset language,


i.e. 𝜓:𝐺→𝐺/Ker(𝜙)

with 𝜓(𝑔)=𝑔𝐾 for 𝐾=ker(𝜙)




If this were settled, then 𝜃 is an isomorphism by the isomorphism theorem.

 

[WWGD]

Please use ## to edit your Latex, it is not rendering anything understandable.

 

[fresh_42]

The basic idea can be seen in your solution, but it's written in an unpleasant way. E.g. you shouldn't use $O$ as a matrix, since it looks like $0$. Try to sort your thoughts: statement - deductions - conclusion.
Have a look at:
https://www.physicsforums.com/insights/how-most-proofs-are-structured-and-how-to-write-them/
And for the use of LaTeX see:
https://www.physicsforums.com/help/latexhelp/
What do $O(3)$ and $SO(3)$ mean?
This means: which of several possible definitions do you use?

Then consider $\det$ and explain, why it is a group homomorphism.
What is its image?
Why is it surjective?
What is its kernel?

Conclude the statement as an application of the isomorphism theorem.