Remark 3

To calculate the invariant factors of a torsion module whose elementary divisors are known, we again lean on the fact that, if the $a_i$ are pairwise coprime elements of $A$, then the product $\prod A/(a_i)$ is a cyclic module isomorphic to $A/(a_1 a_2 ... a_n)$ ($VII$, p. 3, Prop. 4). Let us illustrate the method by looking at the example of the group $G(464\ 600) = M$: write the elementary divisors $p^n$ of $M$ which are powers of the same irreducible $p$ on the same line, beginning with those of greatest exponent; extend these lines to lines of equal length by putting in 1's where necessary:

Then the invariant factors are the products of elements in the same column: $1100, 20, 2, 2, 2$. Indeed $M$ is isomorphic to a product of cyclic groups of orders 1100, 20, 2, 2, 2 by Prop. 4 of $VII$, p. 3; since each of these orders is a multiple of the next, these are the invariant factors of $M$ ($VII$, p. 22, Th. 3).