On Submodularity and K$\mathfrak{F}$-Subnormality in Finite Groups

Abstract

Let $\mathfrak{F}$ be a formation, and let $G$ be a finite group. A subgroup $H$ of $G$ is called $\mathrm{K}\mathfrak{F}$-subnormal (submodular) in $G$ if there is a subgroup chain $H=H_{0}\leq H_{1}\leq\mathinner{\ldotp\ldotp\ldotp}\leq H_{n-1}\leq H_{n}=G$ such that, for every $i$ either $H_{i}$ is normal in $H_{i+1}$ or $H_{i+1}^{\mathfrak{F}}\leq H_{i}$ ($H_{i}$ is a modular subgroup of $H_{i+1}$, respectively). We prove that, in a group, a primary subgroup is submodular if and only if it is $\mathrm{K}\mathfrak{U}_{1}$-subnormal. Here $\mathfrak{U}_{1}$ is a formation of all supersolvable groups of square-free exponent. Moreover, for a solvable subgroup-closed formation $\mathfrak{F}$, every solvable $\mathrm{K}\mathfrak{F}$-subnormal subgroup of a group $G$ is contained in the solvable radical of $G$. We also obtain a series of applications of these results to the investigation of groups factorized by $\mathrm{K}\mathfrak{F}$-subnormal and submodular subgroups.