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]
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
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 \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

Template:Finite-join-closed

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.