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 $⟨H,xH{x}^{-1}⟩$.

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.

Title	pronormal subgroup
Canonical name	PronormalSubgroup
Date of creation	2013-03-22 16:28:13
Last modified on	2013-03-22 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