Left-transitively fixed-depth subnormal subgroup

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

For its use outside the wiki, please define the term when using it. If you are aware of an equivalent standard term, please leave a comment on the talk page
VIEW: Definitions built on this | Facts about this: (facts closely related to Left-transitively fixed-depth subnormal subgroup, all facts related to Left-transitively fixed-depth subnormal subgroup) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
Learn more about terminology local to the wiki | view a complete list of such terminology

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]

|Get subgroup property lookup help |Get exploration suggestions
VIEW RELATED: Subgroup property implications | Subgroup property non-implications | Subgroup metaproperty satisfactions | Subgroup metaproperty dissatisfactions | Subgroup property satisfactions |Subgroup property dissatisfactions
VIEW FACTS THAT:use, or start with, a subgroup satisfying the property|prove, or show that, a subgroup satisfies the property

This is a variation of subnormality|Find other variations of subnormality |

Definition

A subgroup $H$ of a group $K$ is termed left-transitively fixed-depth subnormal in $K$ if there exists a natural number $k\geq 1$ 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 $l\geq k$.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Characteristic subgroup	invariant under all automorphisms; for characteristic subgroups, we can set $k=1$	Characteristic of normal implies normal		|FULL LIST, MORE INFO
Left-transitively 2-subnormal subgroup	obtained by setting $k=2$			|FULL LIST, MORE INFO
Cofactorial automorphism-invariant subgroup				|FULL LIST, MORE INFO
Subgroup-cofactorial automorphism-invariant subgroup				|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Subnormal subgroup			Normal not implies left-transitively fixed-depth subnormal	|FULL LIST, MORE INFO

Related properties

Property	Meaning	Proof of one non-implication	Proof of other non-implication	Notions stronger than both	Notions weaker than both
Right-transitively fixed-depth subnormal subgroup				|FULL LIST, MORE INFO	Subnormal subgroup|FULL LIST, MORE INFO

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.
ABOUT THIS PROPERTY: View variations of this property that are transitive | View variations of this property that are not transitive
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 $H\leq K\leq G$ 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).
View other trim subgroup properties | View other trivially true subgroup properties | View other identity-true subgroup properties

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.