Let T be the torsion subgroup of a finitely generated abelian group. Suppose T Z m1

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Let T be the torsion subgroup of a finitely generated abelian group. Suppose T ≈ Zm1 x Zm2 x · · · x Zmr ≈ Zn1, x Zn2 x · · · x Zns, where mi divides mi+1 for i = 1, · · ·, r - 1, and nj divides nj+1 for n = 1, · · ·, s - 1, and m1 > 1 and n1 > 1. We wish to show that r = s and m= nk fork = 1, · · ·, r, demonstrating the uniqueness of the torsion coefficients.

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

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