Archiv der Mathematik

, Volume 83, Issue 3, pp 193–203 | Cite as

Modularity in the lattice of Σ-permutable subgroups

Original Paper


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 UV 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.

Mathematics Subject Classification (2000):

Primary: 20F16 Secondary: 20D20 


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Copyright information

© Birkhäuser Verlag, Basel 2004

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceSUNY MorrisvilleMorrisvilleUSA

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