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.
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