November 7th, 2010, 11:08 AM  #1 
Newbie Joined: Dec 2009 Posts: 7 Thanks: 0  Isomorphic
Suppose m, n are positive integers which are coprime. Show that U_n X U_m is isomorphic to U_mn. What is the function that maps U_n X U_m to U_mn? Thx! 
November 12th, 2010, 11:42 AM  #2 
Site Founder Joined: Nov 2006 From: France Posts: 824 Thanks: 7  Re: Isomorphic
There's not necessarily only one such function. Naturally, such a function maps a generator of U_n x U_m, for instance (e^(2iPi/m),e^(2iPi/n)), onto a generator of U_(mn), for instance e^(2iPi/(mn)). (Proof: use the fact that both sets have same cardinal, thus it is enough to prove that such a function is a group endomorphism and is injective). 

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