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 $\omega$ denotes some nonempty set of primes, and $\tau$ 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 $f:\omega\cup\{\omega'\}\to\{\text{formations of groups}\}$ are called $\omega$-local satellites (formation $\omega$-functions). Such functions are used to study the structure of $\omega$-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 $\tau$, the lattice $l_{\omega_{\infty}}^{\tau}$ of all $\tau$-closed totally $\omega$-saturated formations is algebraic. This generalizes the results by V.G. Safonov. In particular, we show that the lattice $l_{p_{\infty}}^{\tau}$ of all $\tau$-closed totally $p$-saturated formations is algebraic as well as the lattice $l_{\infty}^{\tau}$ of all $\tau$-closed totally saturated formations. Similar results are obtained for lattices of functorially closed totally partially saturated formations corresponding to certain subgroup functors $\tau$. Thus, we find new classes of algebraic lattices of formations of finite groups.