Left-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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Definition

A subgroup $H$ of a group $K$ is termed left-transitively fixed-depth subnormal in $K$ if there exists a natural number $k\ge 1$ such that $H$ is left-transitively $k$-subnormal in $K$. In other words, whenever $K$ is a $k$-subnormal subgroup of a group $G$, $H$ is also $k$-subnormal in $G$.

Note that any subgroup that is left-transitively $k$-subnormal is also left-transitively $l$-subnormal for $l\ge k$.

Relation with other properties

Stronger properties

• Characteristic subgroup: For characteristic subgroups, we can set $k=1$. For full proof, refer: Characteristic of normal implies normal
• Left-transitively 2-subnormal subgroup: Obtained by setting $k=2$. Also related:
  • Cofactorial automorphism-invariant subgroup

Weaker properties

• Subnormal subgroup: For full proof, refer: Normal not implies left-transitively fixed-depth subnormal

Related properties

• Right-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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ABOUT TRANSITIVITY: View a complete list of transitive subgroup properties|View a complete list of facts related to transitivity of subgroup properties |Read a survey article on proving transitivity

If $H\le K\le G$ are such that $H$ is left-transitively $k$-subnormal in $K$ and $K$ is left-transitively $l$-subnormal in $G$, then $H$ is left-transitively $max\{k,l\}$-subnormal in $G$.

Intersection-closedness

This subgroup property is finite-intersection-closed; a finite (nonempty) intersection of subgroups with this property, also has this property
View a complete list of finite-intersection-closed subgroup properties

An intersection of finitely many such subgroups again has the property. In particular, the intersection of a left-transitively $k$-subnormal subgroup and a left-transitively $l$-subnormal subgroup is left-transitively $max\{k,l\}$-subnormal.

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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A join of finitely many such subgroups again has the property. In particular, the join of a left-transitively $k$-subnormal subgroup and a left-transitively $l$-subnormal subgroup is left-transitively $max\{k,l\}$-subnormal.