Elementary divisors of a finite abelian group

Let \(G\) be a finite abelian group. By the elementary divisor decomposition, there are primes \(p_1,\ldots,p_m\), not necessarily distinct, and integers \(\alpha_1,\ldots,\alpha_m\geq 1\) with \[ G \cong \bigoplus_{i=1}^m \ZZ/p_i^{\alpha_i} \ZZ, \] and the list \(p_1^{\alpha_1},\ldots,p_m^{\alpha_m}\) is unique up to order. These prime powers are the elementary divisors of \(G\).