On two open problems of the theory of permutable subgroups of finite groups

Let $\sigma=\{\sigma_{i}|i\in I\}$ be some partition of the set of all primes $\mathbb{P}$, $G$ a finite group and $\sigma(G)=\{\sigma_{i}|\sigma_{i}\cap\pi(G)\ne\emptyset\}$. A set $\mathcal{H}$ of subgroups of $G$ is said to be a complete Hall $\sigma$-set of $G$ if every member $\ne 1$ of $\mathcal{H}$ is a Hall $\sigma_{i}$-subgroup of $G$ for some $\sigma_{i}\in\sigma$ and $\mathcal{H}$ contains exactly one Hall $\sigma_{i}$-subgroup of $G$ for every $\sigma_{i}\in\sigma(G)$; $G$ is said to be $\sigma$-full if $G$ possesses a complete Hall $\sigma$-set. A subgroup $A$ of $G$ is said to be $\sigma$-permutable in $G$ if $G$ possesses a complete Hall $\sigma$-set and $A$ permutes with each Hall $\sigma_{i}$-subgroup $H$ of $G$, that is, $AH=HA$ for all $i\in I$. We prove that if $G$ is $\sigma$-full, then the set $\mathcal{L}_{\sigma\operatorname{per}}(G)$, of all $\sigma$-permutable subgroups of $G$, forms a sublattice of the lattice of all subgroups of $G$. Also, answering to [9, Question 6.13], we describe the conditions under which the lattice $\mathcal{L}_{\sigma\operatorname{per}}(G)$ is distributive.