Algebraicity of lattice of $\tau$-closed totally $\omega$-saturated formations of finite groups

Abstract

All groups considered in this paper are assumed to be finite. The symbol denotes some nonempty set of primes, and is a subgroup functor in the sense of A.N. Skiba. We recall that a formation is a class of groups that is closed under taking homomorphic images and finite subdirect products. Functions of the form : ∪ { ′ } → {formations of groups} are called-local satellites (formation-functions). Such functions are used to study the structure of-saturated formations. The paper is devoted to studying the properties of the lattice of all closed functorially totally partially saturated formations related to the algebraicity concept for a lattice of formations. We prove that for each subgroup functor , the lattice ∞ of all-closed totally-saturated formations is algebraic. This generalizes the results by V.G. Safonov. In particular, we show that the lattice ∞ of all-closed totally-saturated formations is algebraic as well as the lattice ∞ of all-closed totally saturated formations. Similar results are obtained for lattices of functorially closed totally partially saturated formations corresponding to certain subgroup functors. Thus, we find new classes of algebraic lattices of formations of finite groups.