Proving Every Element in G is a Square Using Factor Groups

[Benzoate]

Homework Statement

Suppose that G is an Abelian group and H is a subgroup of G. If every element of H is a square and every element of G/H is a square , prove that every element of G is a square.

Homework Equations

The Attempt at a Solution

Let a and b be elements of G. The ab=ba since G is an abeleian group. If H is a subgroup of G, then doesn't H share the same opperations with G? If so, since every element of in H is a square, then a^2*b^2 =(aa)(bb)=(bb)(aa) since G is abelian, H should be abelian. Therefore , there is an element in G that is a square

 

[CompuChip]

If H is a subgroup of G, then indeed the operation on H is the same as that from G. And since G is abelian, so is H. Indeed, there is an element of G that is a square (in fact, any element from G that is in H is a square).
But the question was to prove that every element of G is a square.

You didn't use the information about G/H yet. So let a be any element of G. Now you will want to prove that there exists some element b (or you could very suggestively name it $\sqrt{a}$ such that b b = a. How can you do this?