pronormal subgroup
A subgroup^{} $H$ of a group $G$ is called a pronormal subgroup if for all $x\in G$ the subgroups $H$ and $xH{x}^{1}$ are conjugate in $\u27e8H,xH{x}^{1}\u27e9$.
Some facts about pronormal subgroups:

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Normal subgroups^{} are pronormal.

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Maximal subgroups are pronormal.

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Abnormal subgroups are pronormal.

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Sylow subgroups of finite groups^{} are pronormal.

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The normalizer^{} of a pronormal subgroup is abnormal.

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A pronormal subgroup is normal if and only if it is subnormal.
Title  pronormal subgroup 

Canonical name  PronormalSubgroup 
Date of creation  20130322 16:28:13 
Last modified on  20130322 16:28:13 
Owner  yark (2760) 
Last modified by  yark (2760) 
Numerical id  6 
Author  yark (2760) 
Entry type  Definition 
Classification  msc 20E99 
Defines  pronormal 
Defines  pronormality 