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September 2nd, 2011, 12:39 AM   #1
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Category-theory (finite group theory) prove

Let G be a group, , and for define by . Let be the category of groups and homomorphisms. Prove is a -representation of G with kernel . is the representation by conjugation of G on H. If H=G, the image of G under is the inner automorphism group of G and is denoted by Inn(G). Prove . (Define Out(G)=Aut(G)/Inn(g) to be the outer automorphism group of G.

I would be really appreciate if you could help me.
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September 2nd, 2011, 04:04 AM   #2
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Re: Category-theory (finite group theory) prove

Well, where did you get stuck?

This is an exercise that mainly consists of understanding the concepts. After that you simply check the axioms. But that are one-line proofs without any fancy tricks.
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September 2nd, 2011, 09:17 AM   #3
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Re: Category-theory (finite group theory) prove

To tell the truth, I didn't get stuck, unfortunatelly I can't start with.
I would appreciate your help!
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September 3rd, 2011, 01:33 AM   #4
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Re: Category-theory (finite group theory) prove

Ok, let's start with the representation. I would say a representation of a group G on an Object O of a category C is a Homomorphism i.e. a Homomorphism. In your case the object is H in the category of groups and you have to show that is a homomorphism. For that you have to check two things:
1)\pi_g is in fact a Group_automorphism of H (i.e., the described map actually goes into )
2)if h is an other element of G then (i.e. the described map is a homomorphism.)
Once you've done that you should compute the kernel as the centralizer. But that should basically consist of writing down the definition of the map and the centralizer.

Do you understand that and can you do 1 and 2? There is really not much to do!
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September 3rd, 2011, 05:58 AM   #5
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Re: Category-theory (finite group theory) prove

Thank you for your help.

I am not sure, if i am right, but I try...
1) is true because is the conjugation of g on H and H is a normal subgroup of G (which is invariant under conjugation by members of the group).
2) is true because <==

The elements of the centralizer are , and why is it the same as the kernel?

Why is it true that ?

I am really grateful for your help!
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September 3rd, 2011, 06:47 AM   #6
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Re: Category-theory (finite group theory) prove

Quote:
Originally Posted by butabi
Thank you for your help.

I am not sure, if i am right, but I try...
1) is true because is the conjugation of g on H and H is a normal subgroup of G (which is invariant under conjugation by members of the group).
2) is true because <==
Good. There where some brackets missing to put gh both in the exponent. But apart from the typo it is correct.

Quote:
The elements of the centralizer are , and why is it the same as the kernel?
Well what is the kernel? The elements that go to 1 in the "Aut" group. What is the one in the Automorphisms? It is the identity. So what does it mean for some g that the conjugation with g on H is the identity map? Well, we need for all x in H that , thus we are back at the definition of the centralizer.

Quote:
Why is it true that ?
You have to show for all Automorphisms that is again an inner automorphism, i.e., the conjugation by some element g' in G. Write down, what the conjugation is, and use that and are homomorphisms and you will see, what element that is.
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September 3rd, 2011, 08:02 AM   #7
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Re: Category-theory (finite group theory) prove

Thank you again. I am very grateful indeed.

So I don't know if I am right:
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September 3rd, 2011, 12:03 PM   #8
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Re: Category-theory (finite group theory) prove

Quote:
Originally Posted by butabi
So I don't know if I am right:
Not quite, my notation was meant to indicate, that is conjugated by :



But you certainly have got the point now.
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September 3rd, 2011, 01:52 PM   #9
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Re: Category-theory (finite group theory) prove

Yeah, sorry, I missunderstood your notation.
Thank you very much for your help once again, you helped me a lot!
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