Pronormal subgroup

In

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.,
  MR 1169099

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