Vol. 60, No. 2, 1975

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On the lattice of normal subgroups of a direct product

Michael Douglas Miller

Vol. 60 (1975), No. 2, 153–158

Suzuki has determined that if G is a direct product G = Πi=1kGi of groups Gi1, 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(ij). 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.

Mathematical Subject Classification
Primary: 20F30
Received: 3 September 1974
Revised: 14 October 1974
Published: 1 October 1975
Michael Douglas Miller