## direct product

Let k be a positive integer.

define G_k = {x| 1<= x <= k with gcd(x,k)=1}

prove that:
a)G_k is a group under multiplication modulos k (i can do that).

b)G_nm = G_n x G_m be defining an isomorphism.

 PhysOrg.com science news on PhysOrg.com >> Hong Kong launches first electric taxis>> Morocco to harness the wind in energy hunt>> Galaxy's Ring of Fire
 Recognitions: Homework Help Science Advisor What have you done for b)? There is only one possible way you can think of to write out a map from G_nm to G_n x G_m, so prove it is an isomorphism. Remember, G_n x G_m looks like pars (x,y)....
 We can use the Chinese Remainder Theorem on this one. Define the mapping, $$\phi: G_{nm}\mapsto G_n\times G_m$$ As, $$\phi(x) = (x\bmod{n} , x\bmod{m})$$ 1)The homomorphism part is trivial. 2)The bijection part is covered by Chinese Remainder Theorem.

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## direct product

but the point is to prove that theorem.

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