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.