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 such that H = H. (Here H denotes the conjugate subgroup gHg.)
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
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