Let \(G\) be a finitely generated abelian group, and for \(n\geq 1\) let \(C_n\) denote the cyclic group of order \(n\). An invariant factor decomposition of \(G\) is an isomorphism \[ G \cong \ZZ^r \times \prod_{k=1}^m C_{n_k} \] with \(r,m\geq 0\) and integers \(n_1,\ldots,n_m\geq 2\) such that \(n_1 \divides n_2\divides \cdots \divides n_m\). The integers \(n_1,\ldots,n_m\) are the invariant factors of \(G\).
Every finitely generated abelian group \(G\) has an invariant factor decomposition, the direct product of the free abelian group \(\ZZ^r\) and finitely many finite cyclic groups. The integers \(r\) and \(m\) and the invariant factors \(n_1,\ldots,n_m\) are uniquely determined by \(G\).