# Thread: Elements of torsion n of a group G

1. ## Elements of torsion n of a group G

Hello, community! I've been struggling forever with this problem. Let G be abelian and G(n) denote the set of elements g of G such that g^n=e. I'm asked to prove G(n) is a subgroup which is okay, nothing really special about it. However, I'm also asked to prove that given any prime p, it is true that |G(p)|^2 ≥ |G(p^2)|, where the bars help denote the number of elements in each group. Thanks for any help!

Pd: G is also finite.

2. ## Re: Elements of torsion n of a group G

Well, it is easy to show that $G(p)$ is a subgroup of $G(p^2)$. I would guess some application of the Lagrange theorem would be appropriate. Assume that $|G(p^2)|>|G(p)|^2$ and find some contradiction, maybe. What have you tried?

3. ## Re: Elements of torsion n of a group G

This is easy if you know the "fundamental theorem of finite abelian groups" -- you can find this theorem and proofs thereof on the net. I've been trying to prove this without the fundamental theorem, but don't see how to do it. If you can't solve the problem with the theorem, post again and I'll try to help.