Pronormal subgroup

In mathematics, especially in the field of group theory, a pronormal subgroup is a subgroup that is embedded in a nice way. Pronormality is a simultaneous generalization of both normal subgroups and abnormal subgroups such as Sylow subgroups,.

A subgroup is pronormal if each of its conjugates is conjugate to it already in the subgroup generated by it and its conjugate. That is, H is pronormal in G if for every g in G, there is some k in the subgroup generated by H and Hg such that Hk = Hg. (Here Hg denotes the conjugate subgroup gHg-1.)

Here are some relations with other subgroup properties:


 * Every normal subgroup is pronormal.
 * Every Sylow subgroup is pronormal.
 * Every pronormal subnormal subgroup is normal.
 * Every abnormal subgroup is pronormal.


 * Every pronormal subgroup is weakly pronormal, that is, it has the Frattini property.
 * Every pronormal subgroup is paranormal, and hence polynormal.