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

Abstract

A subgroup $H$ of a finite group $G$ is called $\mathfrak{U}$-subnormal in Kegel’s sense or $K-\mathfrak{U}$-subnormal in $G$ if there exists a chain of subgroups $H = H_0 \leq H_1 \leq . . . \leq H_t = G$ such that either $H_{i-1}$ is normal in $H_i$ or $H_i/(H_{i-1})H_i$ is supersoluble for any $i = 1, . . . , t$. We describe finite groups for which every 2-maximal or every 3-maximal subgroup is $K-\mathfrak{U}$-subnormal.