How to show SU(n) is a normal subgroup of U(n)

In summary, the conversation discusses the concept of normal subgroups and the proof that SU(n) is a normal subgroup of U(n). The conversation also addresses the question of whether the musings constitute a formal proof and whether it is necessary to prove certain statements or if they can be assumed based on existing theorems. The final suggestion is to clarify with a teacher for the best approach in an exam situation.
  • #1
vertices
62
0
Hi

I'd like to show that SU(n) is a normal subgroup of U(n).

Here are my thoughts:

1)The kernel of of homomorphism is a normal subgroup.

2)So if we consider a mapping F: G-> G'=det(G)

3)Then all elements of G which are SU(n), map to the the identity of G', therefore SU(n) is a normal subgroup of GL(n,C).

Firstly, is this correct?

Also, a very stupid question, but can I ask whether my musings constitute a formal 'proof'? I mean, would I need to show prove statements (1) and (2) (ie. show it obeys the homom. property) or can I just say that I've assumed that they are for the purposes of the proof at hand?

Thanks.
 
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  • #2
vertices said:
Hi

I'd like to show that SU(n) is a normal subgroup of U(n).

Here are my thoughts:

1)The kernel of of homomorphism is a normal subgroup.

2)So if we consider a mapping F: G-> G'=det(G)

3)Then all elements of G which are SU(n), map to the the identity of G', therefore SU(n) is a normal subgroup of GL(n,C).

Firstly, is this correct?

Also, a very stupid question, but can I ask whether my musings constitute a formal 'proof'? I mean, would I need to show prove statements (1) and (2) (ie. show it obeys the homom. property) or can I just say that I've assumed that they are for the purposes of the proof at hand?

Thanks.

you are correct - formal proofs can assume other known theorems - it is a matter of how much your reader needs to see and how much he already knows.
 
  • #3
wofsy said:
you are correct - formal proofs can assume other known theorems - it is a matter of how much your reader needs to see and how much he already knows.

thanks wofsy... I should have given a bit of context.

I'm going to sit an exam on introductory group theory, lie algebras and representations very soon... I feel a bit unsure when I see "prove that" questions (my excuse, if you can call it that, is that my background isn't in mathematics but I am doing a masters in one and I don't know many of the things I am expected to know...)
 
  • #4
vertices said:
thanks wofsy... I should have given a bit of context.

I'm going to sit an exam on introductory group theory, lie algebras and representations very soon... I feel a bit unsure when I see "prove that" questions (my excuse, if you can call it that, is that my background isn't in mathematics but I am doing a masters in one and I don't know many of the things I am expected to know...)

I would say that theorems in your book or theorems proved in class you can refer to without proof. But the best way to know is to ask you teacher.
 

1. How do you define a normal subgroup?

A normal subgroup is a subgroup of a group that is invariant under conjugation by elements of the larger group. In other words, for any element in the normal subgroup, conjugation by any element in the larger group will result in another element in the normal subgroup.

2. What is the significance of proving that SU(n) is a normal subgroup of U(n)?

Proving that SU(n) is a normal subgroup of U(n) is important because it helps establish the structure and properties of these groups. It also allows for the use of quotient groups and cosets, which are useful tools in group theory and algebraic geometry.

3. How can you show that SU(n) is a subgroup of U(n)?

To show that SU(n) is a subgroup of U(n), we need to prove that it satisfies the three criteria of a subgroup: closure, identity element, and inverse elements. This can be done by showing that the product of any two elements in SU(n) is also in SU(n), that the identity matrix is in SU(n), and that the inverse of any element in SU(n) is also in SU(n).

4. Can you explain the relationship between SU(n) and U(n)?

SU(n) and U(n) are both special unitary groups, which are subgroups of the general linear group GL(n,C). The difference between them is that SU(n) consists of matrices with determinant 1, while U(n) consists of matrices with complex determinant of absolute value 1. In other words, SU(n) is a subgroup of U(n) since it is a subset of matrices with determinant 1.

5. What are some applications of understanding the normal subgroup SU(n) in U(n)?

Understanding the normal subgroup SU(n) in U(n) has applications in many areas, including quantum mechanics, quantum information theory, and mathematical physics. It is also important in the study of Lie groups and their representations, which have applications in diverse fields such as differential geometry, topology, and number theory.

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