July 3rd, 2011, 12:50 PM  #1 
Newbie Joined: Jul 2011 Posts: 2 Thanks: 0  Isomorphism
Hi everyone! I have an abelian group which is a finit group of order n and a function and I want to prove that if then is an isomorphism. Someone told me first prove that is an homomorphism and then prove that for some but I don't mind how to do that. So...some suggestions? Thanks! 
July 3rd, 2011, 01:11 PM  #2 
Newbie Joined: Jul 2011 Posts: 2 Thanks: 0  Re: Isomorphism
I forgot to said that 
July 8th, 2011, 02:58 AM  #3 
Member Joined: Jul 2011 From: Trieste but ever Naples in my heart! Italy, UE. Posts: 62 Thanks: 0 
Have you noted that being an abelian groups than is a his endomorphism? 
July 9th, 2011, 04:18 PM  #4 
Senior Member Joined: Oct 2009 Posts: 105 Thanks: 0  Re: Isomorphism
For the Isomorphism part, you have if then This is what you want to prove. Use the fact that any element (other than the identy) raised to the mth power, is not the identity. The rest of the Isomorphism part should follow after that. 
July 10th, 2011, 12:57 PM  #5 
Senior Member Joined: Jul 2011 Posts: 227 Thanks: 0  Re: Isomorphism
If you can prove your homomorphism is a surjection and a bijection that will implicate it's a bijection and so an isomorphism. And isomorphism i: GG is called an automorphism. 
July 11th, 2011, 02:24 AM  #6  
Global Moderator Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 4  Re: Isomorphism Quote:
(I think he meant "injection")  
July 11th, 2011, 06:00 AM  #7  
Senior Member Joined: Jul 2011 Posts: 227 Thanks: 0  Re: Isomorphism Quote:
 

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