1. ## Abelian Groups

Prove that a finite, simple, Abelian group must be isomorphic to Z_p for some prime p.

I've started a proof, but am getting stuck. I've written..

Let G be a finite, simple, Abelian group. Choose a nonzero element a E G. Consider the set Zg={pg, p E Z}, a nonzero subgroup of G. Since G is finite, it is isomorphic to Zp for some p...

after this, im not sure how to get to the point that p is prime. help please?!

2. What if $|G|$ is composite? Can you say anything about its subgroups?

3. i'm sorry, i'm not sure what you mean by that..

i'm trying to find a correspondence between subgroups of Zn and divisors of n to show that if Zn is simple, then n must be a prime number

4. Suppose that the order of $G$ is composite, i.e. it has some nontrivial divisors. Then, it is possible to produce a proper subgroup, with order of one of the divisors. (How?) Is there anything special about that subgroup that would cause a contradiction?