|
|
|
|
permutable subgroup
|
(Definition)
|
|
|
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 $H\per G$ to indicate that $H$ is a permutable subgroup of $G$ .
Permutable subgroups were introduced by Øystein Ore, 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.
- 1
- Stewart E. Stonehewer, Permutable subgroups of infinite groups, Math. Z. 125 (1972), 1-16. (This paper is available from GDZ.)
|
"permutable subgroup" is owned by yark.
|
|
(view preamble | get metadata)
| Other names: |
quasinormal subgroup, quasi-normal subgroup |
| Also defines: |
permutable, quasinormal, quasi-normal |
|
|
Cross-references: subnormal, finitely generated group, ascendant, normal subgroups, subgroup, group
There are 5 references to this entry.
This is version 6 of permutable subgroup, born on 2006-09-17, modified 2008-11-23.
Object id is 8372, canonical name is PermutableSubgroup.
Accessed 4271 times total.
Classification:
| AMS MSC: | 20E07 (Group theory and generalizations :: Structure and classification of infinite or finite groups :: Subgroup theorems; subgroup growth) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
