Pronormal subgroup
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In mathematics, in the field of group theory, a pronormal subgroup is a subgroup such that every conjugate subgroup to it is a conjugate subgroup to it in the subgroup generated by them. That is, H is pronormal in G if, for every 
, we have: H and Hx are conjugates in the subgroup generated by H and Hx.
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 pronormal subgroup is weakly pronormal, that is, it has the Frattini property
- Every pronormal subgroup is paranormal, and hence polynormal
- Every abnormal subgroup is pronormal
