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.

Characterize m_{r - 1} and n_{s -1}, showing that they are equal, and continue to show m_{r - i} = n_{s -1} for i = 1, · · ·, r - 1, and then r = s.