On sublattices of the subgroup lattice defined by formation Fitting sets

By A. Skiba. 2020

54 citations recorded by the source corpus.

Abstract Throughout this paper, G always denotes a group and L ( G ) is the lattice of all subgroups of G. If K ⊴ H ≤ G , then H / K is called a section of G; such a section is called normal if K , H ⊴ G . We call any set Σ of normal sections of G a stratification of G provided: (i) Σ is G-closed, that is, H / K ∈ Σ whenever H / K ≃ G T / L ∈ Σ , and (ii) L / K , H / L ∈ Σ for each triple K L H , where H / K ∈ Σ and L ⊴ G . Now let Σ be any stratification of G. Then we write L Σ ( G ) to denote the set of all subgroups A of G such that A G / A G ∈ Σ . We say that a stratification Σ of G is a formation Fitting set of G provided: (i) H / ( K ∩ N ) ∈ Σ for every two sections H / K , H / N ∈ Σ , and (ii) H V / K ∈ Σ for every two sections H / K , V / K ∈ Σ . We prove that the set L Σ ( G ) forms a sublattice of L ( G ) for every formation Fitting set Σ of G and we also discuss some applications of sublattices of this kind.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/ea60378395d4c3e50e3fa1347e2c924de65a9d99