Laws of the Lattices of $\sigma$-Local Formations of Finite Groups

Abstract

Let n be a positive integer, and let $\sigma = \{\sigma _i \mid i \in I\}$ be a partition of the set of all primes. It is shown that every law of the lattice of all formations is fulfilled in the lattice of all n-multiply $\sigma$-local formations of finite groups. This result implies that the lattice of all n-multiply $\sigma$-local formations of finite groups is modular but not distributive for any nonnegative integer n. Let ${\mathfrak {F}}$ and ${\mathfrak {H}}$ be n-multiply $\sigma$-local formations of finite groups such that ${\mathfrak {H}} \subseteq {\mathfrak {F}}$. Denote by ${\mathfrak {F}} /_n^\sigma {\mathfrak {H}}$ the lattice of all n-multiply $\sigma$-local formations ${\mathfrak {M}}$ such that ${\mathfrak {H}} \subseteq {\mathfrak {M}} \subseteq {\mathfrak {F}}$. If ${\mathfrak {M}} \subset {\mathfrak {F}}$ and the lattice ${\mathfrak {F}} /_n^\sigma {\mathfrak {M}}$ consists of only two elements then ${\mathfrak {M}}$ is called a maximal n-multiply $\sigma$-local subformation of ${\mathfrak {F}}$. The properties of the intersection of the maximal n-multiply $\sigma$-local subformations of $\mathfrak F$ are described.