# direct product

by barbiemathgurl
Tags: product
 P: 12 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.
 Sci Advisor HW Helper P: 9,398 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)....
 P: 291 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.