PlanetMath: permutable subgroup

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permutable subgroup

(Definition)

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

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.

References

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.

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Other names:

quasinormal subgroup, quasi-normal subgroup

Also defines:

permutable, quasinormal, quasi-normal

Attachments:: example of non-permutable subgroup (Example) by Algeboy

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Cross-references: subnormal, finitely generated group, ascendant, normal subgroups, subgroup, group
There are 5 references to this entry.

This is version 5 of permutable subgroup, born on 2006-09-17, modified 2006-09-18.
Object id is 8372, canonical name is PermutableSubgroup.
Accessed 2476 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

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