On one generalization of modular subgroups

Abstract

Let $G$ be a finite group. If $M_n<M_{n-1}<\dots <M_1<M_0=G$ where $M_i$ is a maximal subgroup of $M_{i-1}$ for all $i=1,\dots ,n$, then $M_n$ ($n>0$) is an \emph{$n$-maximal subgroup} of $G$. A subgroup $M$ of $G$ is called \emph{modular} if the following conditions are held: (i) $⟨X,M\cap Z⟩=⟨X,M⟩\cap Z$ for all $X\le G,Z\le G$ such that $X\le Z$, and (ii) $⟨M,Y\cap Z⟩=⟨M,Y⟩\cap Z$ for all $Y\le G,Z\le G$ such that $M\le Z$. In this paper, we study finite groups whose $n$-maximal subgroups are modular.

13 pages