On σ-inductive lattices of n-multiply σ-local formations of finite groups

Abstract

Throughout this paper, all groups are finite and [Formula: see text] always denotes a group. Let [Formula: see text] be some partition of the set of all primes [Formula: see text], [Formula: see text] be a class of groups, [Formula: see text], and [Formula: see text] A function [Formula: see text] of the form [Formula: see text] is called a formation [Formula: see text]-function. For any formation [Formula: see text]-function [Formula: see text] the class [Formula: see text] is defined as follows: [Formula: see text] If for some formation [Formula: see text]-function [Formula: see text] we have [Formula: see text] then the class [Formula: see text] is called [Formula: see text]-local and [Formula: see text] is called a [Formula: see text]-local definition of [Formula: see text] If, in addition, [Formula: see text] for all [Formula: see text], then [Formula: see text] is called integrated. Every formation is called 0-multiply [Formula: see text]-local. For [Formula: see text] a formation [Formula: see text] is called [Formula: see text]-multiply [Formula: see text]-local provided either [Formula: see text] is the class of all identity groups or [Formula: see text] where [Formula: see text] is [Formula: see text]-multiply [Formula: see text]-local for all [Formula: see text] Let [Formula: see text] be a set of subgroups of [Formula: see text] such that [Formula: see text]. Then [Formula: see text] is called a subgroup functor if for every epimorphism [Formula: see text] : [Formula: see text] and any groups [Formula: see text] and [Formula: see text] we have [Formula: see text] and [Formula: see text]. A class of groups [Formula: see text] is called [Formula: see text]-closed if [Formula: see text] for all [Formula: see text]. Let [Formula: see text] be a complete lattice of formation. Then the symbol [Formula: see text] denotes the set of all formations [Formula: see text] such that [Formula: see text] where [Formula: see text] for all [Formula: see text] A complete lattice [Formula: see text] we call [Formula: see text]-inductive, if for any set [Formula: see text] and for any set [Formula: see text], where [Formula: see text], [Formula: see text] is an integrated, and [Formula: see text] for all [Formula: see text] we have [Formula: see text] where [Formula: see text] is the intersection of all formations from [Formula: see text] containing [Formula: see text] and [Formula: see text] is a formation [Formula: see text]-function such that for all [Formula: see text] [Formula: see text] is the intersection of all formations from [Formula: see text] containing [Formula: see text]. We prove that the set of all [Formula: see text]-closed [Formula: see text]-multiply [Formula: see text]-local formations forms a complete [Formula: see text]-inductive lattice of formations.