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September 2nd, 2011, 12:39 AM  #1 
Member Joined: Sep 2010 Posts: 60 Thanks: 0  Categorytheory (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. 
September 2nd, 2011, 04:04 AM  #2 
Senior Member Joined: Sep 2008 Posts: 150 Thanks: 5  Re: Categorytheory (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 oneline proofs without any fancy tricks. 
September 2nd, 2011, 09:17 AM  #3 
Member Joined: Sep 2010 Posts: 60 Thanks: 0  Re: Categorytheory (finite group theory) prove
To tell the truth, I didn't get stuck, unfortunatelly I can't start with. I would appreciate your help! 
September 3rd, 2011, 01:33 AM  #4 
Senior Member Joined: Sep 2008 Posts: 150 Thanks: 5  Re: Categorytheory (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! 
September 3rd, 2011, 05:58 AM  #5 
Member Joined: Sep 2010 Posts: 60 Thanks: 0  Re: Categorytheory (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! 
September 3rd, 2011, 06:47 AM  #6  
Senior Member Joined: Sep 2008 Posts: 150 Thanks: 5  Re: Categorytheory (finite group theory) prove Quote:
Quote:
Quote:
 
September 3rd, 2011, 08:02 AM  #7 
Member Joined: Sep 2010 Posts: 60 Thanks: 0  Re: Categorytheory (finite group theory) prove
Thank you again. I am very grateful indeed. So I don't know if I am right: 
September 3rd, 2011, 12:03 PM  #8  
Senior Member Joined: Sep 2008 Posts: 150 Thanks: 5  Re: Categorytheory (finite group theory) prove Quote:
But you certainly have got the point now.  
September 3rd, 2011, 01:52 PM  #9 
Member Joined: Sep 2010 Posts: 60 Thanks: 0  Re: Categorytheory (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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