Left-transitively fixed-depth subnormal subgroup

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
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]
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 $k \ge 1$ 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

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 Left-transitively 2-subnormal subgroup|FULL LIST, MORE INFO
Left-transitively 2-subnormal subgroup obtained by setting $k = 2$ |FULL LIST, MORE INFO
Cofactorial automorphism-invariant subgroup Left-transitively 2-subnormal subgroup|FULL LIST, MORE INFO
Subgroup-cofactorial automorphism-invariant subgroup Left-transitively 2-subnormal 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

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 $H \le K \le G$ 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).
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.