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.

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