On Normal Subgroups of Product of Groups

The object of this paper is to find a necessary and sufficient condition for the groups $G_1,G_2,\mathrm{.}\mathrm{.}\mathrm{.},G_n$ so that every normal subgroup of the product ${\prod }_{i=1}^n{G}_i$ is of the type ${\prod }_{i=1}^n{N}_i$ with $N_i \trianglelefteq G_i$,$i=1,2, …, n$. As a consequence we obtain a well-known result due to R. Remak about centreless completely reducible groups having finitely many direct factors.