Paranormal subgroup

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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This is a variation of normality|Find other variations of normality | Read a survey article on varying normality

Definition

Definition with symbols

A subgroup $H$ of a group $G$ is termed paranormal if for any $g\in G$ , there exists $x\in <H,H^{g}>$ such that $<H,H^{x}>=<H,H^{g}>$ .

Relation with other properties

Stronger properties

Weaker properties

Polynormal subgroup

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.
ABOUT THIS PROPERTY: View variations of this property satisfying intermediate subgroup condition | View variations of this property not satisfying intermediate subgroup condition
ABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition

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