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

Let $\mathfrak F$ be a formation, and let $G$ be a finite group. A subgroup $H$ of $G$ is called \lb $\mathrm{K}\mathfrak F$‑subnormal (submodular) in $G$ if there is a subgroup chain $H=H_0\le H_1 \le \ldots \le H_{n-1}\le H_n=G$ such that, for every $i$ either $H_{i}$ is normal in $H_{i+1}$ or $H_{i+1}^\mathfrak{F} \le H_i$ ($H_i$ 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.