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Pronormal subgroup information


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 Hg such that Hk = Hg. (Here Hg 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.

and 7 Related for: Pronormal subgroup information

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Pronormal subgroup

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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...

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Normal subgroup

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Descendant subgroup Quasinormal subgroup Seminormal subgroup Conjugate permutable subgroup Modular subgroup Pronormal subgroup Paranormal subgroup Polynormal...

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Weakly normal subgroup

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N_{G}(H)} , we have g ∈ N G ( H ) {\displaystyle g\in N_{G}(H)} . Every pronormal subgroup is weakly normal. Ballester-Bolinches, Adolfo; Esteban-Romero, R....

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Abnormal subgroup

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weakly abnormal subgroup is a self-normalizing subgroup. Every abnormal subgroup is a pronormal subgroup, and hence a weakly pronormal subgroup, a paranormal...

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Paranormal subgroup

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subgroup properties: Every pronormal subgroup, and hence, every normal subgroup and every abnormal subgroup, is paranormal. Every paranormal subgroup...

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Subnormal subgroup

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quasinormal subgroup, and, more generally, every conjugate-permutable subgroup, of a finite group is subnormal. Every pronormal subgroup that is also...

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Polynormal subgroup

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subgroup properties: Every weakly pronormal subgroup is polynormal. Every paranormal subgroup is polynormal. "Polynormal subgroup - Groupprops". groupprops.subwiki...

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