 [SOLVED] Group theory problem  

Author  Message 

Bruno Admin
Posts : 184 Join date : 20090915 Age : 31 Location : the infinite, frictionless plane of uniform density
 Subject: [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 

 
Mohammad Descartes
Posts : 100 Join date : 20091105 Age : 34 Location : Right behind you
 Subject: 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 

 
Bruno Admin
Posts : 184 Join date : 20090915 Age : 31 Location : the infinite, frictionless plane of uniform density
 Subject: 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. 

 
Mohammad Descartes
Posts : 100 Join date : 20091105 Age : 34 Location : Right behind you
 Subject: Re: [SOLVED] Group theory problem Thu Nov 12, 2009 12:56 am  
 O.K, More details For n=p^{\alpha} ,where p is a prime number and \alpha is a natural number, the claim is obvious 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 

 
Bruno Admin
Posts : 184 Join date : 20090915 Age : 31 Location : the infinite, frictionless plane of uniform density
 Subject: Re: [SOLVED] Group theory problem Thu Nov 12, 2009 11:45 am  
 Good! 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? 

 
Mohammad Descartes
Posts : 100 Join date : 20091105 Age : 34 Location : Right behind you
 Subject: Re: [SOLVED] Group theory problem Thu Nov 12, 2009 11:11 pm  
 

 
Bruno Admin
Posts : 184 Join date : 20090915 Age : 31 Location : the infinite, frictionless plane of uniform density
 Subject: Re: [SOLVED] Group theory problem Fri Nov 13, 2009 8:08 pm  
 Awesome! Thanks 

 
Mohammad Descartes
Posts : 100 Join date : 20091105 Age : 34 Location : Right behind you
 Subject: Re: [SOLVED] Group theory problem Fri Nov 13, 2009 8:31 pm  
 

 
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 Subject: Re: [SOLVED] Group theory problem  
 

 
 [SOLVED] Group theory problem  
