Abstract
All groups under consideration are finite. Let be some partition of the set of all primes , G be a group, be a class of groups, , and A function f of the form is called a formation σ-function. For any formation σ-function f the class is defined as follows: If for some formation σ-function f we have then the class is called σ-local and f is called a σ-local definition of Every formation is called 0-multiply -local. For a formation is called n-multiply σ-local provided either is the class of all identity groups or where is -multiply σ-local for all Let be a set of subgroups of G such that . Then τ is called a subgroup functor if for every epimorphism : and any groups and we have and . A formation of groups is called τ-closed if for all . A complete lattice of formations θ is called separable, if for any term signatures , any θ-formations and any group there are groups such that . We prove that the lattice of all τ-closed n-multiply σ-local formations is a separable lattice of formations.
Acknowledgments
The author is deeply grateful to the referee for useful suggestions.
Disclosure statement
The author declares the absence of a conflict of interest.