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, (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 H^g such that H^k = H^g. (Here H^g 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.

References

• Doerk, Klaus; Hawkes, Trevor (1992), Finite soluble groups, de Gruyter Expositions in Mathematics, vol. 4, Berlin: Walter de Gruyter & Co., ISBN 978-3-11-012892-5, MR 1169099