# permutable subgroup

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

## References

• 1 Stewart E. Stonehewer, Permutable subgroups of infinite groups, Math. Z. 125 (1972), 1–16. (This paper is http://gdz.sub.uni-goettingen.de/dms/resolveppn/?GDZPPN002410435available from GDZ.)
Title permutable subgroup PermutableSubgroup 2013-03-22 16:15:47 2013-03-22 16:15:47 yark (2760) yark (2760) 9 yark (2760) Definition msc 20E07 quasinormal subgroup quasi-normal subgroup permutable quasinormal quasi-normal