ON INDUCTIVE LATTICES OF SATURATED FORMATIONS

Abstract

All groups considered are finite. The symbol Fp(G) denotes the p-nilpotent radical of a group G; p(G) is the set of primes dividing the order of G. Let f be a function of the form f : ℙ → {formations of groups}. (Ú) We consider the class of groups LF ( f ) = (G | G / Fp (G) ∈ f (p) for all p ∈ p(G)). If F is a formation such that F = LF ( f ) for
a function f of the form (Ú), then F is said to be saturated and f is said to be a local satellite of F. Let F be a saturated formation. We write F /l F ∩ N to denote the lattice of all saturated formations lying between F and F ∩ N, where N is the class of all
nilpotent groups. Let Q be a complete lattice of formations. Then we denote by Ql the set of all formations having a local Q-valued satellite. A satellite f is called Q-valued if all values of f belong to Q. Let X ⊆ F ∈ Q be a collection of group. We write QformX
to denote the intersection of all formations of Θ containing all groups of X. Let {Fi | i ∈ I} be an arbitrary collection of formations in Θ. We denote ∨Q (Fi | i ∈ I) = Qform i
i∈I        F . Let { fi | i ∈ I } be a collection of Θ-valued satellites. Then ∨Q ( fi | i ∈ I ) denotes the satellite f such that ( ) form i ( ) i I f p f p ∈   = Q      for every p ∈ ℙ. A complete lattice Ql is called inductive (see Skiba A. N. Algebra formacij [Algebra of Formations]. Minsk, Belaruskaja navuka Publ., 1997) if for any collection {Fi = LF ( fi ) | i ∈ I }, where fi is an integrated satellite of Fi ∈ Ql, the following equality holds: ( | ) ( ( | )) l i i i I LF f i I Q Q ∨∈ = ∨ ∈ F . In this paper, we prove the following
T h e o r e m. Let F be a saturated formation. Then the lattice F /l F ∩ N is inductive.