On $\tau$-closed $n$-multiply $\sigma$-local formations of finite groups

Abstract

All groups under consideration are finite. Let $\sigma =\{{\sigma }_i\mid i\in I\}$ be some partition of the set of $\mathbb{P}$, $G$ be a group, and $\mathfrak{F}$ be a class of groups. Then $\sigma (G)=\{{\sigma }_i\mid {\sigma }_i\cap \pi (G)\ne \varnothing \}$ and $\sigma (\mathfrak{F})={\cup }_{G\in \mathfrak{F}}\sigma (G).$ A function $f$ of the form $f:\sigma \to \{\text{formations of groups}\}$ is called a formation $\sigma$-function. For any formation $\sigma$-function $f$ the class $L{F}_{\sigma }(f)$ is defined as follows: $$L{F}_{\sigma }(f)=(G\text{ is a group }\mid G=1\text{ or }G\ne 1\text{ and }G/O_{{\sigma }_i^{\prime },{\sigma }_i}(G)\in f({\sigma }_i)\text{ for all }{\sigma }_i\in \sigma (G)).$$ If for some formation $\sigma$-function $f$ we have $\mathfrak{F}=L{F}_{\sigma }(f),$ then $\mathfrak{F}$ is called $\sigma$-local, $f$ is called a $\sigma$-local definition of $\mathfrak{F}.$ Every formation is called 0-multiply $\sigma$-local. For $n>0,$ a formation $\mathfrak{F}$ is called $n$-multiply $\sigma$-local provided either $\mathfrak{F}=(1)$ or $\mathfrak{F}=L{F}_{\sigma }(f),$ where $f({\sigma }_i)$ is $(n-1)$-multiply $\sigma$-local for all ${\sigma }_i\in \sigma (\mathfrak{F}).$ Let $\tau (G)$ be a set of subgroups of $G$ such that $G\in \tau (G)$. Then $\tau$ is called a subgroup functor if for every epimorphism $\phi$ : $A\to B$ and any groups $H\in \tau (A)$ and $T\in \tau (B)$ we have $H^{\phi }\in \tau (B)$ and $T^{{\phi }^{-1}}\in \tau (A)$. A class $\mathfrak{F}$ is called $\tau$-closed if $\tau (G)\subseteq \mathfrak{F}$ for all $G\in \mathfrak{F}$. We describe some properties of $\tau$-closed $n$-multiply $\sigma$-local formations, as well as we prove that the set $l_{{\sigma }_n}^{\tau }$ of all $\tau$-closed $n$-multiply $\sigma$-local formations forms a complete modular algebraic lattice. In addition, we proof that $l_{{\sigma }_n}^{\tau }$ is $\sigma$-inductive and $\mathfrak{G}$-separable.