Let \(G\) be a finitely generated abelian group, and for \(n\geq 1\) let \(C_n\) denote the cyclic group of order \(n\). An elementary divisor decomposition of \(G\) is an isomorphism \[ G \cong \ZZ^r \times \prod_{k=1}^m C_{p_k^{e_k}} \] with \(r,m\geq 0\), primes \(p_1,\ldots,p_m\) that need not be distinct, and exponents \(e_1,\ldots,e_m\geq 1\). The prime powers \(p_1^{e_1},\ldots,p_m^{e_m}\) are the elementary divisors of \(G\).
Every finitely generated abelian group \(G\) has an elementary divisor decomposition. The integer \(r\) and the list \(p_1^{e_1},\ldots,p_m^{e_m}\), up to reordering, are uniquely determined by \(G\).