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November 7th, 2010, 10:08 AM   #1
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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?

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November 12th, 2010, 10:42 AM   #2
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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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