Subgroup of a Quotient is a Quotient of a Subgroup

[Szichedelic]

Homework Statement

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."

Homework Equations

See below.

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.

 

[R136a1]

I think your approach is fine.

 

[Office_Shredder]

The question is literally asking you to prove part of the fourth isomorphism theorem, so if you're allowed to cite the relevant part then you're done, but I would double check the context of the question (when it came up in the class, what it says on the homework assignment elsewhere) to see if you're allowed to use that theorem or are supposed to be proving the fourth isomorphism theorem.