Finite groups with modular and submodular subgroups

Abstract

A subgroup $H$ of a group $G$ is modular in $G$ if $H$ is a modular element of subgroup lattice of $G$, and is submodular in $G$ if there is a subgroup chain $H=H_0\leq\ldots\leq H_i\leq H_{i+1}\leq \ldots \leq H_n=G$ such that $H_i$ is modular in $H_{i+1}$ for every $i$. We prove that if every Sylow subgroup of a group $G$ is modular in $G$, then $G$ is supersolvable and $G/F(G)$ is a cyclic group of square-free order. We also obtain new signs of supersolvabilty of groups with some submodular subgroups (normalizers of Sylow subgroups, Hall subgroups, maximal subgroups). For a such group $G$, $G/\Phi(G)$ is a supersolvable group of square-free exponent. Moreover, we describe the structure of groups with modular (submodular) or self-normalizing primary subgroups.