Problem:
Kernel and Homomorphism
If A and B are subsets of a group G, define
AB = {ab|a2A,b2B}. Now suppose phi: G -> G0 is a homomorphism of groups and N = Ker(phi) is its kernel.
(i) If H is a subgroup of G, show that HN = NH. (Warning: this is an equation of sets; proceed accordingly; do not assume that G is abelian.)
(ii) Show that phi-inverse[phi[H]] = HN.