On weakly $N$-subnormal subgroups of finite groups

Abstract

Let $G$ be a finite group and $A,N\le G$. Then $A_{\mathrm{sn}G}$ is the subnormal core of $A$ in $G$, that is, the subgroup of $A$ generated by all subnormal subgroups of $G$, contained in $A$; $A^{\mathrm{sn}G}$ is the subnormal closure of $A$ in $G$, that is, the intersection of all subnormal subgroups of $G$ containing $A$. We say that a subgroup $A$ of $G$ is (i) $N$-subnormal in $G$ if $N\cap A^{\mathrm{sn}G}=N\cap A_{\mathrm{sn}G}$; (ii) weakly $N$-subnormal in $G$ if for some subnormal subgroup $T$ of $G$ we have $AT=G$ and $A\cap T\le S\le A$, where $S$ is $N$-subnormal in $G$. In this paper, we consider some applications of these two concepts. In particular, we prove that a finite group $G$ is soluble if and only if $G$ has a normal subgroup $N$ with soluble factor $G/N$ such that in each maximal chain $M_3<M_2<M_1<M_0=G$ of $G$ of length $3$, at least one of the subgroups $M_3$, $M_2$ or $M_1$ is weakly $N$-subnormal in $G$.