Polynormal subgroup

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

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Definition

Definition with symbols

A subgroup $H$ of a group $G$ is termed polynormal if given any $g\in G$, there exists a $x\in H^{<g>}$ such that $H^{<x>}=H^{<g>}$. Here $H^{<g>}$ denotes the smallest subgroup of $G$ containing $H$, which is closed under conjugation by $g$.

Relation with other properties

Stronger properties

• Normal subgroup
• Maximal subgroup
• Abnormal subgroup
• Pronormal subgroup
• Weakly abnormal subgroup
• Weakly pronormal subgroup
• Paranormal subgroup
• Sylow subgroup in a finite group

Weaker properties

• Fan subgroup
• Intermediately subnormal-to-normal subgroup: For full proof, refer: Polynormal implies intermediately subnormal-to-normal

Metaproperties

Intermediate subgroup condition

YES: This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.
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If $H$ is polynormal in $G$, $H$ is also polynormal in any intermediate subgroup $K$.

Trimness

This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself).
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The whole group and the trivial subgroup are polynormal; in fact they are normal.

References

• On the arrangement of intermediate subgroups by M. S. Ba and Z. I. Borevich
• On the arrangement of subgroups by Z. I. Borevich, Zap. Nauchn. Semin. tOMI, 94, 5-12 (1979)
• On the lattice of subgroups by Z. I. Borevich and O. N. Macedonska, Zap. Nauchn. Semin. LOMI, 103, 13-19, 1980
• Testing of subgroups of a finite group for some embedding properties like pronormality by V. I. Mysovskikh