Remark

While the orders $n_1, \ldots, n_q$ of the cyclic groups of which F is the direct sum are well defined by the divisibility condition of Th. 3, it is not the same for the groups themselves: for example, in the product G of $\mathbf{Z}/(p)$ with itself (for p prime), the subgroups are precisely the $\mathbf{F}_p$-vector subspaces, and G is the direct sum of two 1-dimensional subspaces in $p(p+1)$ different ways.