Lattice characterizations of soluble and supersoluble finite groups

Let $G$ be a finite group and ${\mathscr L}_{sn}(G)$ be the lattice of all subnormal subgroups of $G$. Let $A$ and $N$ be subgroups of $G$ and $1, G\in {\mathscr L}$ be a sublattice of ${\mathscr L}_{sn}(G)$, that is, $A\cap B$, $\langle A, B \rangle \in {\mathscr L}$ for all $A, B \in {\mathscr L} \subseteq {\mathscr L}_{sn}(G)$. Then: $A^{{\mathscr L}}$ is the $\mathscr L$-closure of $A$ in $G$, that is, the intersection of all subgroups in $ {\mathscr L}$ containing $A$ and $A_{{\mathscr L}}$ is the $\mathscr L$-core of $A$ in $G$, that is, the subgroup of $A$ generated by all subgroups of $A$ belonging $\mathscr L$. We say that $A$ is an $N$-${\mathscr L}$-subgroup of $G$ if either $A\in {\mathscr L}$ or $A_{{\mathscr L}} < A < A^{\mathscr L}$ and $N$ avoids every composition factor $H/K$ of $G$ between $A_{{\mathscr L}}$ and $ A^{\mathscr 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.