Products of quotient groups

Suppose G is a nontrivial finite group and H,KG are normal subgroups with gcd(|H|,|K|)=1.

  1. Define a nontrivial group homomorphism ϕ:GG/H×G/K
  2. Prove G is isomorphic to a subgroup of G/H×G/K.
  3. Suppose gcd(m,n)=1. Prove ZmnZm×Zn.