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 [SOLVED] Group theory problem

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Bruno
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PostSubject: [SOLVED] Group theory problem   Wed Nov 11, 2009 5:46 pm



Last edited by Bruno on Fri Nov 13, 2009 7:53 pm; edited 1 time in total
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Mohammad
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PostSubject: Re: [SOLVED] Group theory problem   Wed Nov 11, 2009 11:44 pm

The only automorphisms can be of the forms A(x)=X^m for some natural numbers m, now when is it an automorphism? Obviously when (m,n)=1, otherwise the order of elements don't agrees but an automorphism has to preserve the order bounce
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Bruno
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PostSubject: Re: [SOLVED] Group theory problem   Thu Nov 12, 2009 12:24 am

Mohammad wrote:
The only automorphisms can be of the forms A(x)=X^m for some natural numbers m

Why? Prove this directly.



Quote :
Obviously when (m,n)=1


Why? Give a proof.

The problem isn't trivial, I don't think you can get rid of it in two lines. pirat
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Mohammad
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PostSubject: Re: [SOLVED] Group theory problem   Thu Nov 12, 2009 12:56 am

O.K, More details Exclamation
For n=p^{\alpha} ,where p is a prime number and \alpha is a natural number, the claim is obvious Smile
Now every finitely generated abelian group G is isomorphic to a direct sum of primary cyclic groups and infinite cyclic groups, in our case we don't need "infinite cyclic groups" so... we are done bounce
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Bruno
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PostSubject: Re: [SOLVED] Group theory problem   Thu Nov 12, 2009 11:45 am

Good! cheers
It's obvious when n=p^alpha and p is an odd prime, because in that case the group of units of Z/nZ is cyclic. But when n=2^alpha?
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Mohammad
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PostSubject: Re: [SOLVED] Group theory problem   Thu Nov 12, 2009 11:11 pm

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PostSubject: Re: [SOLVED] Group theory problem   Fri Nov 13, 2009 8:08 pm

Awesome! Thanks bom
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PostSubject: Re: [SOLVED] Group theory problem   Fri Nov 13, 2009 8:31 pm

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