(b) Let H be a subgroup of Z n , and let H = {x Z n | k N: kx H} (Here, N denotes the natural numbers, i., the strictly positive integers.) Show that...

(b) Let H be a subgroup of Z n , and let H¯ = {x ∈ Z n | ∃k ∈ N: kx ∈ H} (Here, N denotes the natural numbers, i.e., the strictly positive integers.) Show that H¯ is a subgroup of Z n , that [H¯ : H] is finite, and that there exists a subgroup K of Z n with Z n = H¯ ⊕ K