# Finite groups with systems of $K$-$\frak{F}$-subnormal subgroups

## TL;DR

This paper studies the structure of finite groups with specific subgroup chains called $K$-$rak{F}$-subnormal subgroups, exploring how these influence the group's overall properties within the framework of $K$-lattice formations.

## Contribution

It introduces new applications of $K$-lattice formations, proving conditions under which a group's structure is constrained by its $K$-$rak{F}$-subnormal subgroups.

## Key findings

- If all $rak{F}$-critical subgroups are $K$-$rak{F}$-subnormal, then $G/F(G)$ belongs to $rak{F}$.
- If all Schmidt subgroups are $K$-$rak{F}$-subnormal, then $G/G_{rak{F}}$ is abelian.
- The results connect subgroup properties with the global structure of finite groups.

## Abstract

Let $\frak {F}$ be a class of group. A subgroup $A$ of a finite group $G$ is said to be $K$-$\mathfrak{F}$-subnormal in $G$ if there is a subgroup chain $$A=A_{0} \leq A_{1} \leq \cdots \leq A_{n}=G$$ such that either $A_{i-1} \trianglelefteq A_{i}$ or $A_{i}/(A_{i-1})_{A_{i}} \in \mathfrak{F}$ for all $i=1, \ldots , n$. A formation $\frak {F}$ is said to be $K$-lattice provided in every finite group $G$ the set of all its $K$-$\mathfrak{F}$-subnormal subgroups forms a sublattice of the lattice of all subgroups of $G$. In this paper we consider some new applications of the theory of $K$-lattice formations. In particular, we prove the following Theorem A. Let $\mathfrak{F}$ be a hereditary $K$-lattice saturated formation containing all nilpotent groups. (i) If every $\mathfrak{F}$-critical subgroup $H$ of $G$ is $K$-$\mathfrak{F}$-subnormal in $G$ with $H/F(H)\in {\mathfrak{F}}$, then $G/F(G)\in {\mathfrak{F}}$. (ii) If every Schmidt subgroup of $G$ is $K$-$\mathfrak{F}$-subnormal in $G$, then $G/G_{\mathfrak{F}}$ is abelian.