Let H be a subgroup of a group G. Prove that

H is a normal subgroup G if and only if Hg = gH for every g that exist G:

(Note: Hg = {hg : h exist in Hg} and gH = {gh : h exist in Hg} are sets. So, in the proof of -->, you suppose that H is a normal subgroup of G and then prove two inclusions: Hg is a subset of gH and gH is a subset of Hg.)