Covering of the orthogonal group

In summary, the conversation discussed the canonical homomorphism in the context of the group 𝑂(3) and its subgroup 𝑆𝑂(3). It was shown that the determinant function πœ™ is a group homomorphism from 𝑂(3) to the group β„€2, with an image of {βˆ’1,1}. The kernel of πœ™ was determined to be 𝑆𝑂(3), since the determinant of a matrix in 𝑆𝑂(3) is always 1. This led to the conclusion that πœ™ is a homomorphism and πœ“ is an isomorphism, as per the isomorphism theorem. Further clarification
  • #1
Jason Bennett
49
3
Homework Statement
see title
Relevant Equations
see below
Progress:πœ™:𝑂(3)β†’β„€2πœ“:𝑂(3)→𝑆𝑂(3)πœƒ:𝑂(3)/𝑆𝑂(3)β†’β„€2
πœ™(𝑂)=det(𝑂)

with π‘‚βˆˆπ‘‚(3), that way

πœ™(𝑂)↦{βˆ’1,1}β‰…β„€2,

where 1 is the identity element.Ker(πœ™) = {π‘‚βˆˆπ‘†π‘‚(3)|πœ™(𝑂)=1}=𝑆𝑂(3), since det(𝑂)=1 for π‘‚βˆˆπ‘†π‘‚(3).By the multiplicative property of the determinant function, πœ™ = homomorphism.
***What is the form of the canonical homomorphism (πœ“) in this case?I'm used to the coset language,i.e. πœ“:𝐺→𝐺/Ker(πœ™)

with πœ“(𝑔)=𝑔𝐾 for 𝐾=ker(πœ™)

If this were settled, then πœƒ is an isomorphism by the isomorphism theorem.
 
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  • #2
Please use ## to edit your Latex, it is not rendering anything understandable.
 
  • #3
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.
 

Related to Covering of the orthogonal group

1. What is the orthogonal group?

The orthogonal group, denoted as O(n), is a mathematical group that consists of all the n x n real orthogonal matrices. These matrices have a special property that the columns and rows are orthonormal vectors, meaning they are perpendicular to each other and have a magnitude of 1.

2. What is the covering of the orthogonal group?

The covering of the orthogonal group, denoted as SO(n), is a special subgroup of O(n) that contains all the proper rotations in n-dimensional space. It is a double cover, meaning each element in SO(n) corresponds to exactly two elements in O(n).

3. Why is the covering of the orthogonal group important?

The covering of the orthogonal group is important because it allows us to study rotations in a more efficient way. Since rotations in n-dimensional space can be represented by elements in SO(n), this subgroup provides a more manageable and intuitive way to analyze and understand rotations.

4. How is the covering of the orthogonal group related to Lie groups?

The covering of the orthogonal group is a special type of Lie group, which is a group that is also a smooth manifold. In this case, SO(n) is a Lie group of dimension n(n-1)/2. This connection to Lie groups allows us to use powerful mathematical tools and techniques to study the properties of SO(n).

5. Can the covering of the orthogonal group be extended to other types of groups?

Yes, the idea of a covering group can be extended to other types of groups, such as the special unitary group SU(n) and the symplectic group Sp(n). These groups also have double covers that are important in various areas of mathematics and physics.

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