permutable subgroup


permutable subgroupsPlanetmathPlanetmath

Let G be a group. A subgroupMathworldPlanetmathPlanetmath H of G is said to be permutable if it permutes with all subgroups of G, that is, KH=HK for all KG. We sometimes write HperG to indicate that H is a permutable subgroup of G.

Permutable subgroups were introduced by Øystein Ore (, who called them quasinormal subgroups.

Normal subgroupsMathworldPlanetmath 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.


  • 1 Stewart E. Stonehewer, Permutable subgroups of infinite groups, Math. Z. 125 (1972), 1–16. (This paper is from GDZ.)
Title permutable subgroup
Canonical name PermutableSubgroup
Date of creation 2013-03-22 16:15:47
Last modified on 2013-03-22 16:15:47
Owner yark (2760)
Last modified by yark (2760)
Numerical id 9
Author yark (2760)
Entry type Definition
Classification msc 20E07
Synonym quasinormal subgroup
Synonym quasi-normal subgroup
Defines permutable
Defines quasinormal
Defines quasi-normal