.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 .
This well may be false: you're only given , not that n is the maximal power of p dividing the order of G.
I'd rather go: let by Sylow theorems there exists . Now, it's easy to prove that any p-group of
order has a normal sbgp. of order for any (by induction, say) . In the present case normality is for free since G is abelian, and still
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?