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

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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ABOUT JOIN-CLOSEDNESS: View all join-closed subgroup properties (or, strongly join-closed properties) | View all subgroup properties that are not join-closed | Read a survey article on proving join-closedness | Read a survey article on disproving join-closedness

In fact, an arbitrary, possibly empty, join of polynormal subgroups is polynormal. For full proof, refer: Polynormality is strongly join-closed

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