Let p be a prime. Let G be a finite abelian group such that xp=e for all X in G.
1) Let k be a positive integer, and let a1,...,ak be elements of G such that the products a1i1 ... akik (the 1 and k on the i's should be subscripts) are distinct for all choices of integers i1 ... ik 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 Z2 X Z2 X Z2 isomorphic to Z2 X Z2.
3) Let G be the direct product of Zp X ... X Zp of m copies of Zp 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 Zp X ... X Zp for an integer k <= (less than or equal to) m. Use Exercise 1.