Prove that the following is a subgroup

Let p be a prime. Let G be a finite abelian group such that x^{p}=e for all X in G.

1) Let k be a positive integer, and let a_{1},...,a_{k} be elements of G such that the products a_{1i}^{1} ... a_{ki}^{k (the 1 and k on the i's should be subscripts)} are distinct for all choices of integers i_{1} ... i_{k} from 0 through p-1. Prove that the set of elements (a1...ak) is a subgroup of G.

2) Prove that

(0,0,0), (1,1,0), (1,0,1), (0,1,1)

is a subgroup of Z_{2} X Z_{2} X Z_{2} isomorphic to Z_{2} X Z_{2}.

3) Let G be the direct product of Z_{p} X ... X Z_{p} of m copies of Z_{p} for a positive integer m. Let H be a subgroup of G that has more than one element. Prove that H is isomorphic to the direct product Z_{p} X ... X Z_{p} for an integer k <= (less than or equal to) m. Use Exercise 1.

Re: Prove that the following is a subgroup

Hey TimsBobby2.

It's been a little while since I did this stuff, but can you show us what you've tried, thought about, and any attempts (both partial and full)?