1. The problem statement, all variables and given/known data I'm trying to prove the statement "Show that a subgroup of a quotient of G is also a quotient of a subgroup of G." 2. Relevant equations See below. 3. The attempt at a solution Let G be a group and N be a normal subgroup of G. Let H be a subgroup of the quotient G/N. Then, by the fourth isomorphism theorem, H is isomorphic to the subgroup of G of the form H/N. I'm wondering if I am done at this point... My teacher's statement is somewhat vague and I can't decide if by "quotient of a subgroup of G," he means G quotiented by ANOTHER normal subgroup or a quotient OF a subgroup of G. If it is the latter, than this problem is trivially true by the fourth isomorphism theorem.