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 $g \in G$, $H$ is a contranormal subgroup in the subgroup $H^{\langle g \rangle}$, i.e., the closure of $H$ under the action by conjugation of the cyclic subgroup generated by $g$.

Relation with other properties

Stronger properties

• Normal subgroup
• Maximal subgroup
• Abnormal subgroup
• Pronormal subgroup
• Weakly abnormal subgroup
• Weakly pronormal subgroup
• Strongly paranormal subgroup
• Paranormal subgroup
• Strongly polynormal 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 $K$. For full proof, refer: Polynormality satisfies intermediate subgroup condition

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.

Join-closedness

YES: This subgroup property is join-closed: an arbitrary (nonempty) join of subgroups with this property, also has this property.
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In fact, an arbitrary, possibly empty, join of polynormal subgroups is polynormal. For full proof, refer: Polynormality is strongly join-closed

Testing

GAP code

One can write code to test this subgroup property in GAP (Groups, Algorithms and Programming), though there is no direct command for it.
View the GAP code for testing this subgroup property at: IsPolynormal
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GAP-codable subgroup property

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