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

Abstract

Let $\mathfrak{F}$ 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 $$A=A_0\le A_1\le \cdots \le A_n=G$$ such that either $A_{i-1}⊴A_i$ or $A_i/(A_{i-1}{)}_{A_i}\in \mathfrak{F}$ for all $i=1,\dots ,n$. A formation $\mathfrak{F}$ 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.

11 pages