On two sublattices of the subgroup lattice of a finite group

Abstract

Abstract Let 𝔉 {\mathfrak{F}} be a non-empty class of groups, let G be a finite group and let ℒ ⁢ ( G ) {\mathcal{L}(G)} be the lattice of all subgroups of G. A chief H / K {H/K} factor of G is 𝔉 {\mathfrak{F}} -central in G if ( H / K ) ⋊ ( G / C G ⁢ ( H / K ) ) ∈ 𝔉 {(H/K)\rtimes(G/C_{G}(H/K))\in\mathfrak{F}} . Let ℒ c ⁢ 𝔉 ⁢ ( G ) {\mathcal{L}_{c\mathfrak{F}}(G)} be the set of all subgroups A of G such that every chief factor H / K {H/K} of G between A G {A_{G}} and A G {A^{G}} is 𝔉 {\mathfrak{F}} -central in G; ℒ 𝔉 ⁢ ( G ) {\mathcal{L}_{\mathfrak{F}}(G)} denotes the set of all subgroups A of G with A G / A G ∈ 𝔉 {A^{G}/A_{G}\in\mathfrak{F}} . We prove that the set ℒ c ⁢ 𝔉 ⁢ ( G ) {\mathcal{L}_{c\mathfrak{F}}(G)} and, in the case when 𝔉 {\mathfrak{F}} is a Fitting formation, the set ℒ 𝔉 ⁢ ( G ) {\mathcal{L}_{\mathfrak{F}}(G)} are sublattices of the lattice ℒ ⁢ ( G ) {\mathcal{L}(G)} . We also study conditions under which the lattice ℒ c ⁢ 𝔑 ⁢ ( G ) {\mathcal{L}_{c\mathfrak{N}}(G)} and the lattice of all subnormal subgroup of G are modular.