Let T be the torsion subgroup of a finitely generated abelian group. Suppose T ≈ Z_m1 x Z_m2 x · · · x Z_mr ≈ Z_n1, x Z_n2 x · · · x Z_ns, where m_i divides m_i+1 for i = 1, · · ·, r - 1, and n_j divides n_j+1 for n = 1, · · ·, s - 1, and m₁ > 1 and n₁ > 1. We wish to show that r = s and m_k = n_k fork = 1, · · ·, r, demonstrating the uniqueness of the torsion coefficients.

Argue from Exercise 20 that m_r and n_s can both be characterized as follows. Let p₁, · · ·, P_t be the distinct primes dividing |T|, and let p₁^h1, · · ·, p_t^hr be the highest powers of these primes appearing in the (unique)  prime-power decomposition Then m_r = n_s = p₁^h1 p₂^h2 · · · P_t^ht .

Data from Exercise 20

Indicate how a prime-power decomposition can be obtained from a torsion-coefficient decomposition.