Theorem 3

— Every finitely generated abelian group G is a direct sum of its torsion subgroup F (subgroup of elements of finite order in G) und a free abelian group of finite rank p (isomorphic to $\mathbf{Z}^p$). The group F is a direct sum of a finite number of cyclic groups of orders $n_1, n_2, \ldots, n_q$, where the $n_i$ are integers $> 1$, each of which divides the previous one; moreover, the integers $p, q$ and $n_i$ ($1 \leq i \leq q$) are uniquely determined by $G$.