April 18th, 2012, 11:14 PM  #1 
Newbie Joined: Apr 2012 Posts: 16 Thanks: 0  isomorphism
Let G be an abelian group of order . Define by , where . Prove if gcd(m,n) = 1 then is an isomorphism.

April 22nd, 2012, 03:08 AM  #2 
Senior Member Joined: Jul 2011 Posts: 227 Thanks: 0  Re: isomorphism
First, prove that is an homomorphism. Afterwards you need to prove is bijective thus ontoone and onto (surjective). To prove if is onetoone (or injective) you have to show: We know that thus you can write with (note that because of ) Thus There's a theorem which says that if is a finite group then the order (in this case ) is a multiple of the order of every element that means and thus but because , has to be minimal and thus therefore and is ontoone. This is just an attempt so I'm not absolutely sure it's entirely correct, but maybe you can do something with my proof. 
July 2nd, 2012, 03:17 AM  #3 
Newbie Joined: Jul 2012 Posts: 4 Thanks: 0  Re: isomorphism
Note that: (a) for each ; (b) gcd(m,n)=1 is equivalent to the statement, there exist integers k and l such that km+ln=1; (c) to show that is a bijection is to show its kernel is trivial and it is surjective. Since for each it follows that: 1. and so is surjective; 2. If then and and so the kernel of is trivial ( is injective) 

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