Join-transitively subnormal subgroup: Difference between revisions

Latest revision as of 16:21, 13 May 2010

Definition

Symbol-free definition

A subgroup of a group is termed join-transitively subnormal if its join (viz., the subgroup generated) with any subnormal subgroup is again subnormal.

Definition with symbols

A subgroup $H$ of a group $G$ is termed join-transitively subnormal if whenever $K\triangleleft \triangleleft G$ (viz., $K$ is subnormal in $G$ ), the join of subgroups $\langle H,K\rangle$ is subnormal in $G$ .

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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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If the ambient group is a finite group, this property is equivalent to the property: subnormal subgroup
View other properties finitarily equivalent to subnormal subgroup | View other variations of subnormal subgroup |

Formalisms

In terms of the join-transiter

This property is obtained by applying the join-transiter to the property: subnormal subgroup
View other properties obtained by applying the join-transiter

The subgroup property of being join-transitively subnormal is obtained by applying the join-transiter to the subgroup property of being subnormal.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Normal subgroup		Normal implies join-transitively subnormal	Follows from 2-subnormal implies join-transitively subnormal, 2-subnormal not implies normal	\|FULL LIST, MORE INFO
2-subnormal subgroup	Normal subgroup of normal subgroup	2-subnormal implies join-transitively subnormal	Follows from finite implies subnormal join property, and there exist subgroups of arbitrarily large subnormal depth in finite groups	\|FULL LIST, MORE INFO
Linear-bound join-transitively subnormal subgroup	There's a constant $a$ such that the join with any $k$ -subnormal subgroup is subnormal of depth at most $ak$	(by definition)	?	\|FULL LIST, MORE INFO
Polynomial-bound join-transitively subnormal subgroup	There's a polynomial function $p$ such that the join with any $k$ -subnormal subgroup is subnormal of depth at most $p(k)$	(by definition)	?	\|FULL LIST, MORE INFO
Subnormal-permutable subnormal subgroup	subnormal and permutes with all subnormal subgroups	Subnormal-permutable and subnormal implies join-transitively subnormal	finite implies subnormal join property, and examples of subnormal subgroups of finite groups that are not subnormal-permutable	\|FULL LIST, MORE INFO
Permutable subnormal subgroup	subnormal and a permutable subgroup of the whole group	Permutable and subnormal implies join-transitively subnormal	(via subnormal-permutable subnormal)	\|FULL LIST, MORE INFO
Perfect subnormal subgroup	subnormal subgroup and also a perfect group	Perfect subnormal implies join-transitively subnormal	normal implies join-transitively subnormal, examples of normal subgroups that are not perfect	\|FULL LIST, MORE INFO
Subnormal subgroup of finite index	subnormal subgroup that is also a subgroup of finite index	Subnormal of finite index implies join-transitively subnormal		\|FULL LIST, MORE INFO
Subnormal subgroup of finite group	subnormal subgroup of finite group	Finite implies subnormal join property		\|FULL LIST, MORE INFO
Conjugate-join-closed subnormal subgroup	join of any collection of its conjugate subgroups is subnormal	Conjugate-join-closed subnormal implies join-transitively subnormal		\|FULL LIST, MORE INFO
Automorph-join-closed subnormal subgroup	join of any collection of its automorphic subgroups is subnormal	(via conjugate-join-closed)		\|FULL LIST, MORE INFO
Intermediately join-transitively subnormal subgroup	join-transitively subnormal in every intermediate subgroup	(by definition)		\|FULL LIST, MORE INFO

Weaker properties

Property	Proof of implication	Intermediate notions
Subnormal subgroup		\|FULL LIST, MORE INFO
Finite-automorph-join-closed subnormal subgroup	Join-transitively subnormal implies finite-automorph-join-closed subnormal	\|FULL LIST, MORE INFO
Finite-conjugate-join-closed subnormal subgroup	(via finite-automorph-join-closed subnormal subgroup)	\|FULL LIST, MORE INFO

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
Transitive subgroup property	No	normal implies join-transitively subnormal, subnormality is not finite-join-closed	We can have $H\leq K\leq G$ such that $H$ is join-transitively subnormal in $K$ and $K$ is join-transitively subnormal in $G$ , but $H$ is not join-transitively subnormal in $G$ .
Trim subgroup property	Yes		Every group is join-transitively subnormal in itself; the trivial subgroup is join-transitively subnormal.
Finite-intersection-closed subgroup property	Known open problem	intersection problem for join-transitively subnormal subgroups	Given join-transitively subnormal subgroups $H,K$ of a group $G$ , is $H\cap K$ necessarily join-transitively subnormal?
Intermediate subgroup condition	Possibly open problem (see intermediately join-transitively subnormal subgroup)		If $H\leq K\leq G$ such that $H$ is join-transitively subnormal in $G$ , is $H$ necessarily join-transitively subnormal in $K$ .
Finite-join-closed subgroup property	Yes		If $H,K\leq G$ are both join-transitively subnormal in $G$ , then $\langle H,K\rangle$ is also join-transitively 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

Clearly, the whole group is join-transitively subnormal, because its join with any subgroup is the whole group. Also, the trivial subgroup is join-transitively subnormal, because its join with any subnormal subgroup is the same subnormal subgroup.

Join-closedness

YES: This subgroup property is join-closed: an arbitrary (nonempty) join of subgroups with this property, also has this property.
ABOUT THIS PROPERTY: View variations of this property that are join-closed | View variations of this property that are not join-closed
ABOUT JOIN-CLOSEDNESS: View all join-closed subgroup properties (or, strongly join-closed properties) | View all subgroup properties that are not join-closed | Read a survey article on proving join-closedness | Read a survey article on disproving join-closedness

By the general theory of transiters, the join-transiter of any subgroup property is itself a finite-join-closed subgroup property.

@@ Line 1: / Line 1: @@
-{{wikilocal}}
-{{subgroup property}}
-{{finitarily equivalent to|subnormal subgroup}}
 ==Definition==
 ===Symbol-free definition===
-A [[subgroup]] of a [[group]] is termed '''join-transitively subnormal''' if its join (viz the subgroup generated) with any [[subnormal subgroup]] is again [[subnormal subgroup|subnormal]].
+A [[subgroup]] of a [[group]] is termed '''join-transitively subnormal''' if its [[defining ingredient::join of subgroups|join]] (viz., the subgroup generated) with any [[defining ingredient::subnormal subgroup]] is again [[subnormal subgroup|subnormal]].
 ===Definition with symbols===
-A [[subgroup]] <math>H</math> of a group <math>G</math> is termed join-transitively subnormal if whenever <math>K \triangleleft \triangleleft G</math> (viz., <math>K</math> is subnormal in <math>G</math>), the group <math>\langle H,K \rangle</math> is subnormal in <math>G</math>.
+A [[subgroup]] <math>H</math> of a group <math>G</math> is termed '''join-transitively subnormal''' if whenever <math>K \triangleleft \triangleleft G</math> (viz., <math>K</math> is subnormal in <math>G</math>), the [[join of subgroups]] <math>\langle H,K \rangle</math> is subnormal in <math>G</math>.
+{{wikilocal}}
+{{subgroup property}}
+{{finitarily equivalent to|subnormal subgroup}}
 ==Formalisms==
@@ Line 46: / Line 46: @@
 | [[Weaker than::Conjugate-join-closed subnormal subgroup]] || join of any collection of its [[conjugate subgroups]] is subnormal ||[[Conjugate-join-closed subnormal implies join-transitively subnormal]] || || {{intermediate notions short|join-transitively subnormal subgroup|conjugate-join-closed subnormal subgroup}}
 |-
-| [[Weaker than::Automorph-join-closed subnormal subgroup]] || join of any collection of its [[automorphic subgroups]] is subnormal || [[Automorph-join-closed implies join-transitively subnormal]] || || {{intermediate notions short|join-transitively subnormal subgroup|automorph-join-closed subnormal subgroup}}
+| [[Weaker than::Automorph-join-closed subnormal subgroup]] || join of any collection of its [[automorphic subgroups]] is subnormal || (via conjugate-join-closed) || || {{intermediate notions short|join-transitively subnormal subgroup|automorph-join-closed subnormal subgroup}}
 |-
 | [[Weaker than::Intermediately join-transitively subnormal subgroup]] || join-transitively subnormal in every intermediate subgroup || (by definition) || || {{intermediate notions short|join-transitively subnormal subgroup|intermediately join-transitively subnormal subgroup}}
@@ Line 64: / Line 64: @@
 ==Metaproperties==
+{| class="sortable" border="1"
+! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols
+|-
+| [[Dissatisfies metaproperty::Transitive subgroup property]] || No || [[normal implies join-transitively subnormal]], [[subnormality is not finite-join-closed]] || We can have <math>H \le K \le G</math> such that <math>H</math> is join-transitively subnormal in <math>K</math> and <math>K</math> is join-transitively subnormal in <math>G</math>, but <math>H</math> is not join-transitively subnormal in <math>G</math>.
+|-
+| [[Satisfies metaproperty::Trim subgroup property]] || Yes || || Every group is join-transitively subnormal in itself; the trivial subgroup is join-transitively subnormal.
+|-
+| [[Finite-intersection-closed subgroup property]] || Known open problem || [[intersection problem for join-transitively subnormal subgroups]] || Given join-transitively subnormal subgroups <math>H,K</math> of a group <math>G</matH>, is <math>H \cap K</math> necessarily join-transitively subnormal?
+|-
+| [[Intermediate subgroup condition]] || Possibly open problem (see [[intermediately join-transitively subnormal subgroup]]) || || If <math>H \le K \le G</math> such that <math>H</math> is join-transitively subnormal in <math>G</math>, is <math>H</math> necessarily join-transitively subnormal in <math>K</math>.
+|-
+| [[Satisfies metaproperty::Finite-join-closed subgroup property]] || Yes || || If <math>H, K \le G</math> are both join-transitively subnormal in <math>G</math>, then <math>\langle H, K \rangle</math> is also join-transitively subnormal.
+|}
 {{trim}}