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 \(\mathrm{K}\mathfrak{F}\)-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.