Deal with showing the uniqueness of the prime powers appearing in the prime-power decomposition of the torsion subgroup T of a finitely generated abelian group.

Referring to Exercise 16, show that Z_p^r [p] ≈ Z_P for any r ≥ 1 and prime p.

Data from Exercise 16

Let G be any abelian group and let n be any positive integer. Show that G[n] = {x ∈ G | nx = 0} is a subgroup of G. (In multiplicative notation, G[n] = {x ∈ G| xⁿ = e}.)