Abstract:
Let $\sigma =\{\sigma_{i} | i\in I\}$ be a partition of the set of all primes $\Bbb{P}$, $G$ be 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 $\sigma _{i}\in \sigma (G)$. 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 a distributive lattice of ${\sigma}$-permutable subgroups.
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