On separability of the lattice of τ-closed n-multiply σ-local formations

Abstract

All groups under consideration are finite. Let $\sigma =\left\{{\sigma }_i\right|i\in I\}$ be some partition of the set of all primes $\mathbb{P}$, G be a group, $\mathfrak{F}$ be a class of groups, $\sigma (G)=\left\{{\sigma }_i\right|{\sigma }_i\cap \pi (G)\ne \varnothing \}$, and $\sigma (\mathfrak{F})={\cup }_{G\in \mathfrak{F}}\sigma (G).$A function f of the form $f:\sigma \to \left\{\text{formations of groups}\right\}$ is called a formation σ-function. For any formation σ-function f the class $L{F}_{\sigma }(f)$ is defined as follows: $$L{F}_{\sigma }(f)=(G\text{is}\mathrm{a}\text{group |G = 1 or G}\ne 1\text{and}{\text{G/O}}_{{\sigma }_{\mathrm{i}}^{\prime },{\sigma }_{\mathrm{i}}}(\mathrm{G})\in \mathrm{f}({\sigma }_i)\text{for all}{\sigma }_{\mathrm{i}}\in \sigma (\mathrm{G})).$$ If for some formation σ-function f we have $\mathfrak{F}=L{F}_{\sigma }(f),$ then the class $\mathfrak{F}$ is called σ-local and f is called a σ-local definition of $\mathfrak{F}.$ Every formation is called 0-multiply $\sigma$-local. For $n>0,$a formation $\mathfrak{F}$ is called n-multiply σ-local provided either $\mathfrak{F}=(1)$ is the class of all identity groups or $\mathfrak{F}=L{F}_{\sigma }(f),$ where $f({\sigma }_i)$ is $(n-1)$-multiply σ-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 τ 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 formation of groups $\mathfrak{F}$ is called τ-closed if $\tau (G)\subseteq \mathfrak{F}$ for all $G\in \mathfrak{F}$. A complete lattice of formations θ is called separable, if for any term $\nu (x_1,\dots ,x_m)$ signatures $\{\cap ,{\vee }_{\theta }\}$, any θ-formations ${\mathfrak{F}}_1,\dots ,{\mathfrak{F}}_m$ and any group $A\in \nu ({\mathfrak{F}}_1,\dots ,{\mathfrak{F}}_m)$ there are groups $A_1\in {\mathfrak{F}}_1,\dots ,A_m\in {\mathfrak{F}}_m$ such that $A\in \nu (\theta \text{form} A_1,\dots ,\theta \text{form} A_m)$. We prove that the lattice of all τ-closed n-multiply σ-local formations is a separable lattice of formations.

KEYWORDS:

• σ-local formation
• τ-closed n-multiply σ-local formation
• lattice of formations
• subgroup functor
• separable lattice of formations

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 20F17
• 20D10

Acknowledgments

The author is deeply grateful to the referee for useful suggestions.

Disclosure statement

The author declares the absence of a conflict of interest.

Additional information

Funding

This research was supported by the Ministry of Education of the Republic of Belarus (No. 20211328).