Title: On the lattice of the $σ$-permutable subgroups of a finite group,
Abstract: Let $\sigma =\{\sigma_{i} | i\in I\}$ be some partition of the set of all
primes $\Bbb{P}$, $G$ a finite group and $\sigma (G) =\{\sigma_{i}
|\sigma_{i}\cap \pi (G)\ne \emptyset \}$. A set ${\cal H}$ of subgroups of $G$
is said to be a complete Hall $\sigma $-set of $G$ if every member $\ne 1$ of
${\cal H}$ is a Hall $\sigma_{i}$-subgroup of $G$ for some $\sigma_{i}\in
\sigma $ and ${\cal H}$ contains exactly one Hall $\sigma_{i}$-subgroup of $G$
for every $\sigma_{i}\in \sigma (G)$. 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 characterize finite groups with distributive lattice of
the ${\sigma}$-permutable subgroups.