Left-transitively fixed-depth subnormal subgroup: Difference between revisions

Revision as of 19:41, 27 May 2010

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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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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	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 <amth>k = 2</math>		\|FULL LIST, MORE INFO
Cofactorial automorphism-invariant subgroup			\|FULL LIST, MORE INFO

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.
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

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.

@@ Line 13: / Line 13: @@
 ===Stronger properties===
-* [[Weaker than::Characteristic subgroup]]: For characteristic subgroups, we can set <math>k = 1</math>. {{proofat|[[Characteristic of normal implies normal]]}}
+{| class="sortable" border="1"
-* [[Weaker than::Left-transitively 2-subnormal subgroup]]: Obtained by setting <math>k = 2</math>. Also related:
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
-** [[Weaker than::Cofactorial automorphism-invariant subgroup]]
+|-
+| [[Weaker than::Characteristic subgroup]] || invariant under all [[automorphism]]s; for characteristic subgroups, we can set <math>k = 1</math> ||[[Characteristic of normal implies normal]] || || {{intermediate notions short|left-transitively fixed-depth subnormal subgroup|characteristic subgroup}}
+|-
+| [[Weaker than::Left-transitively 2-subnormal subgroup]] || obtained by setting <amth>k = 2</math> || || || {{intermediate notions short|left-transitively fixed-depth subnormal subgroup|left-transitively 2-subnormal subgroup}}
+|-
+| [[Weaker than::Cofactorial automorphism-invariant subgroup]] || || || || {{intermediate notions short|left-transitively fixed-depth subnormal subgroup|cofactorial automorphism-invariant subgroup}}
+|}
 ===Weaker properties===