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

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Discussion Overview

The discussion revolves around the question of whether SU(n) is a normal subgroup of U(n). Participants explore the implications of group homomorphisms and the properties of kernels in relation to subgroup normality. The context includes theoretical aspects of group theory, particularly in preparation for an exam on related topics.

Discussion Character

  • Technical explanation
  • Homework-related

Main Points Raised

  • One participant suggests that the kernel of a homomorphism is a normal subgroup and proposes a mapping from G to G' defined by the determinant.
  • It is noted that elements of SU(n) map to the identity in the context of the determinant, implying that SU(n) is a normal subgroup of GL(n,C).
  • Another participant confirms that formal proofs can assume known theorems, indicating that the depth of explanation may depend on the audience's familiarity with the material.
  • A participant expresses uncertainty about "prove that" questions, citing a lack of background in mathematics despite pursuing a master's degree.
  • There is a suggestion that theorems from textbooks or class can be referenced without proof, but it is advised to consult a teacher for clarity on expectations.

Areas of Agreement / Disagreement

Participants generally agree on the validity of assuming known theorems in formal proofs. However, there is no consensus on the necessity of proving specific statements related to the normality of SU(n) within U(n), and uncertainty remains regarding the participant's confidence in their proof-writing abilities.

Contextual Notes

Participants express varying levels of familiarity with group theory concepts, indicating potential limitations in their understanding of the material required for formal proofs.

Who May Find This Useful

This discussion may be useful for students preparing for exams in group theory, particularly those who are transitioning from different academic backgrounds and seeking clarification on proof-writing conventions.

vertices
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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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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.
 
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...)
 
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.
 

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