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July 3rd, 2011, 01:50 PM   #1
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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?

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July 3rd, 2011, 02:11 PM   #2
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Re: Isomorphism

I forgot to said that
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July 8th, 2011, 03:58 AM   #3
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Have you noted that being an abelian groups than is a his endomorphism?
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July 9th, 2011, 05:18 PM   #4
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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.
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July 10th, 2011, 01:57 PM   #5
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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: G-G is called an automorphism.
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July 11th, 2011, 03:24 AM   #6
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Re: Isomorphism

Quote:
Originally Posted by Siron
If you can prove your homomorphism is a surjection and a bijection that will implicate it's a bijection...
Indeed, (P ^ Q) => Q.

(I think he meant "injection")
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July 11th, 2011, 07:00 AM   #7
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Re: Isomorphism

Quote:
Originally Posted by The Chaz
Quote:
Originally Posted by Siron
If you can prove your homomorphism is a surjection and a bijection that will implicate it's a bijection...
Indeed, (P ^ Q) => Q.

(I think he meant "injection")
Yes, indeed I meant 'injection' .
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