On two sublattices of the subgroup lattice of a finite group

Abstract

Let F be a non-empty class of groups, let G be a finite group and let L.G/ be the lattice of all subgroups of G. A chief H=K factor of G is F-central in G if .H=K/ Ì .G=CG.H=K// 2 F. Let LcF.G/ be the set of all subgroups A of G such that every chief factor H=K of G between AG and AG is F-central in G; LF.G/ denotes the set of all subgroups A of G with AG=AG 2 F. We prove that the set LcF.G/ and, in the case when F is a Fitting formation, the set LF.G/ are sublattices of the lattice L.G/. We also study conditions under which the lattice LcN.G/ and the lattice of all subnormal subgroup of G are modular.