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Isomorphisms, Residue Classes and Multiplication Tables

Show that U(10) is isomorphic to Z_4 and write out the isomorphism explicitly.

I know that U(10) and Z_4 are both cyclic, thus they are ismorphic but for writing out the isomorphism, I need assistance.

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Z_10 = {[0], [1], ... [9]} and the subset
U(10) = {[1], [3], [7], [9]} consists of residue classes [s] mod 10 such that gcd(s, 10) = 1, where we choose representatives s between 1 and ...

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Isomorphisms, Residue Classes and Multiplication Tables are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

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