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 November 7th, 2010, 10:08 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, 10:42 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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### u_mn is isomorphic to u_m u_n

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