permutable subgroup
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≤G.
We sometimes write HperG
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 |
|---|---|
| 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 |