Modularity in the lattice of Σ-permutable subgroups

Abstract.

For a Hall system Σ of a finite solvable group G, it is known that the set $\mathcal{P}(\Sigma )$ of Σ-permutable subgroups is a sublattice of the subgroup lattice of G. We investigate the class SPM of groups in which the lattice $\mathcal{P}(\Sigma )$ is modular. We prove that if $\mathcal{P}(\Sigma )$ is modular, then U⊥ V for all $U,V \in \mathcal{P}(\Sigma )$ (an evidently stronger condition). Both of these phenomena—the modularity of $\mathcal{P}(\Sigma )$ and whether two Σ-permutable subgroups U and V permute with each other—are shown to be determined “locally,” by what happens at each prime. The class SPM is shown to be quotient closed, but not direct product or subgroup closed.