Lattice characterizations of soluble and supersoluble finite groups

Abstract

Let $G$ be a finite group and ${\cal L}_{sn}(G)$ be the lattice of all subnormal subgroups of $G$. Let $A$ and $N$ be subgroups of $G$ and $G\in {\cal L}$ be a sublattice of ${\cal L}_{sn}(G)$, that is, $A\cap B$, $\langle A, B \rangle \in {\cal L}$ for all $A, B \in {\cal L} \subseteq {\cal L}_{sn}(G)$. Then: $A^{{\cal L}}$ is the $\cal L$-closure of $A$ in $G$, that is, the intersection of all subgroups in $ {\cal L}$ containing $A$ and $A_{{\cal L}}$ is the $\cal L$-core of $A$ in $G$, that is, the subgroup of $A$ generated by all subgroups of $A$ belonging $\cal L$. We say that $A$ is an $N$-${\cal L}$-subgroup of $G$ if either $A\in {\cal L}$ or $A_{{\cal L}} < A < A^{\cal L}$ and $N$ avoids every composition factor $H/K$ of $G$ between $A_{{\cal L}}$ and $ A^{\cal L}$, that is, $N\cap H=N\cap K$. Using this concept, we give new characterizations of soluble and supersoluble finite groups. Some know results are extended.