Suzuki has determined that if G
is a direct product G = Πi=1kGi of groups Gi≠1, then the lattice L(G) of
subgroups of G is the direct product of the lattices L(Gi) if and only if the
order of any element in Gi is finite and relatively prime to the order of any
element in Gj(i≠j). An exercise in Zassenhaus’ The Theory of Groups asks the
reader to prove an analogous result for the lattice of normal subgroups. In
§1, we derive this result for the case of the direct product of two groups.
(The generalization to the direct product of any finite number of groups is
straightforward.) In §2, we use results obtained in §1 to study in detail the
normal subgroup lattice of the direct product of finitely many symmetric
groups.
