Right-transitively fixed-depth subnormal subgroup

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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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If the ambient group is a finite group, this property is equivalent to the property: subnormal subgroup
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Definition

Definition with symbols

A subgroup $H$ of a group $G$ is termed right-transitively fixed-depth subnormal if there exists $k\geq 1$ such that $H$ is right-transitively $k$-subnormal in $G$: whenever $K$ is a $k$-subnormal subgroup of $H$, $K$ is also $k$-subnormal in $G$.

Any right-transitively $k$-subnormal subgroup is also right-transitively $l$-subnormal for $l\geq k$.

(Here, a $k$-subnormal subgroup is a subnormal subgroup whose subnormal depth is at most $k$).

Relation with other properties

Stronger properties

• Transitively normal subgroup: Here, $k=1$. Also related:
  • Central subgroup
  • Central factor
  • Direct factor
  • Conjugacy-closed normal subgroup
  • SCAB-subgroup
• Right-transitively 2-subnormal subgroup: Here, $k=2$. Also related:
  • Subgroup of abelian normal subgroup
  • Abelian normal subgroup
  • Base of a wreath product
  • Normal T-subgroup
• Nilpotent subnormal subgroup
• Finite subnormal subgroup

Weaker properties

• Subnormal subgroup

Related properties

• Left-transitively fixed-depth subnormal subgroup

Metaproperties

Transitivity

This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property in the whole group.
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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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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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