Rewrite the theorem in your own words.

What is its usefulness?

How does this

theorem strengthen our understanding of finitely generated abelian groups.

Theorem 2.6. Ler G be a finitely generated abelian group. (1) There is a unique nonnegative integer s such that the number of infinite cyclic summands in any decomposition of G as a direct sum of cyclic groups is precisely s; (ii) either G is free abelian or there is a unique list of (not necessarily distinct) positive integers m...., me such that m > 1, mm,l... me and Gez....zm F with F free abelian: (iii) either G is free abelian or there is a list of positive integers p."....", which is unique except for the order of its members, such that ..... are (not necessarily distinct) primes, s, ..., Sare (not necessarily distinct) positive integers and G=7,...7 F with F free abelian