permutable subgroup

permutable subgroups

Let $G$ be a group.A subgroup $H$ of $G$ is said to be permutableif it permutes with all subgroups of $G$,that is, $KH=HK$ for all $K\le G$.We sometimes write $H\mathrm{per}G$to indicate that $H$ is a permutable subgroup of $G$.

Permutable subgroups were introduced by Øystein Ore (http://planetmath.org/OysteinOre),who called them quasinormal subgroups.

Normal subgroups are clearly permutable.

Permutable subgroups are ascendant.This is a result of Stonehewer[1],who also showed that in a finitely generated group,all permutable subgroups are subnormal.