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. 

Let G be a finitely generated abelian group and Tp the subgroup defined in Exercise 14. Suppose T_p ≈ Z_p^r1 x Z_p^r2 X ... X Z_p^rm ≈ Z_p^s1 Z_p^s2 X ... X z_p^sn, where 1 ≤ r₁ ≤ r₂ ≤ · · · ≤ r_m and 1 ≤ s₁ ≤ s₂ ≤ · · · ≤ s_n, We need to show that m = n and r_i = s_i for i = 1, · · ·, n to complete the demonstration of uniqueness of the prime-power decomposition. 

a. Use Exercise 18 to show that n = m. 

b. Show r₁ = s₁. Suppose r_i = s_i for all i < j. Show r_j = S_j, which will complete the proof.