Remark 1

If $M$ is a finite abelian group, then its structure can be described by writing down its elementary divisors, each repeated as often as its multiplicity. We will say, for example, that $M$ is « of type $(2, 2, 4, 27, 27, 25)$ » (or that it is « a group $(2, 2, 4, 27, 27, 25)$ ») if it is isomorphic to the product of two groups $\mathbf{Z}/(2)$, one group $\mathbf{Z}/(2^2)$, two groups $\mathbf{Z}/(3^3)$ and one group $\mathbf{Z}/(5^2)$.