Upper semimodularity of finite subgroup lattices (Q1916061)

The paper gives a very nice combinatorial characterization of upper semimodular subgroup lattices. On one hand, using the known description of finite groups with upper semimodular subgroup lattices, the author shows that in such a lattice every interval sublattice has the property that the number of atoms is less than or equal to the number of coatoms in it. On the other hand, even a stronger converse is true: by inspecting the minimal sections with non-upper-semimodular subgroup lattice it is obtained that if the subgroup lattice of a finite group is not upper semimodular, then it contains an interval sublattice of height 3 in which the number of atoms exceeds the number of coatoms.