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, (Doerk & Hawkes 1992, I.§6).
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
- Doerk, Klaus; Hawkes, Trevor (1992), Finite soluble groups, de Gruyter Expositions in Mathematics, 4, Berlin: Walter de Gruyter & Co., ISBN 978-3-11-012892-5, MR 1169099
|This abstract algebra-related article is a stub. You can help Wikipedia by expanding it.|