Remark 2

If a finitely generated torsion module $M$ over a principal ideal domain $A$ is given as a direct sum of cyclic modules isomorphic to $A/(a_i)$ (in particular if the invariant factors of $M$ are known), then the elementary divisors of $M$, and their multiplicities, can be determined by noticing that $A/(a)$ is isomorphic to the product of the $A/(p^{n(p)})$, where $a = e \prod_{p \in P} p^{n(p)}$ is the decomposition of $a$ into irreducible factors ($VII$, p. 3). Let us study for example the multiplicative group $G(464\ 600)$, where $G(n)$ denotes the multiplicative group $(\mathbf{Z}/n\mathbf{Z})^*$ ($VII$, p. 12). Since $464\ 600 = 2^3 \cdot 5 \cdot 23 \cdot 101$, this group is isomorphic to the product of the groups $G(2^3)$, $G(5^2)$, $G(23)$ and $G(101)$ ($VII$, p. 13, Th. 3); now the last three groups are cyclic of orders 20, 22 and 100, and $G(2^3)$ is the product of two cyclic groups of order 2 (\emph{loc. cit.}); since $20 = 2^2 \cdot 5$, $22 = 2 \cdot 11$ and $100 = 2' \cdot 5^2$, the group $G(464\ 600)$ is of type $(2, 2, 2, 2^2, 2^2, 2^2, 5, 5^2, 11)$.