PlanetMath: pronormal subgroup

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pronormal subgroup

(Definition)

A subgroup of a group is called a pronormal subgroup if for all $x\in G$ the subgroups and $xHx^{-1}$ are conjugate in ${\left\langle H,xHx^{-1}\right\rangle}$ .

Some facts about pronormal subgroups:

Normal subgroups are pronormal.
Maximal subgroups are pronormal.
Abnormal subgroups are pronormal.
Sylow subgroups of finite groups are pronormal.
The normalizer of a pronormal subgroup is abnormal.
A pronormal subgroup is normal if and only if it is subnormal.

"pronormal subgroup" is owned by yark.

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Also defines:

pronormal, pronormality

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Cross-references: subnormal, abnormal, normalizer, finite groups, Sylow subgroups, abnormal subgroups, maximal subgroups, normal subgroups, conjugate, group, subgroup
There is 1 reference to this entry.

This is version 3 of pronormal subgroup, born on 2006-12-18, modified 2006-12-18.
Object id is 8634, canonical name is PronormalSubgroup.
Accessed 1227 times total.

Classification:

AMS MSC:

20E99 (Group theory and generalizations :: Structure and classification of infinite or finite groups :: Miscellaneous)

Pending Errata and Addenda

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