If is a finite abelian group and is a prime such that divides , then prove that has a subgroup of order .

Note: This proof is very similar to the First Sylow Theorem, but in this one, G is abelian.

Attempt at the proof:

If , then .

Consider a group .

Claim: is a group such that .

I know that I have to use Lagrange's Theorem to show that . I just don't know how to do this. Can anyone help?