Transitively normal subgroup: Difference between revisions

Latest revision as of 15:05, 23 August 2016

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Definition

Equivalent definitions in tabular format

No.	Shorthand	A subgroup of a group is termed transitively normal if ...	A subgroup $H$ of a group $G$ is termed transitively normal if ...
1	transitively normal	every normal subgroup of the subgroup is normal in the whole group.	whenever $K$ is a normal subgroup of $H$ , $K$ is also a normal subgroup of $G$ .
2	normal automorphisms restriction	every normal automorphism of the whole group restricts to a normal automorphism of the subgroup.	for any normal automorphism $σ$ of $G$ , the restriction of $σ$ to $H$ is a normal automorphism of $H$ .
3	normal CEP-subgroup	it is a normal subgroup as well as a CEP-subgroup: every normal subgroup of it is an intersection with it of a normal subgroup of the whole group	$H$ is a normal subgroup of $G$ and for any normal subgroup $K$ of $H$ there exists a normal subgroup $L$ of $G$ such that $K = H \cap L$ .

Equivalence of definitions

Further information: Equivalence of definitions of transitively normal subgroup

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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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 page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: normal subgroup and CEP-subgroup
View other subgroup property conjunctions | view all subgroup properties

This is a variation of normality|Find other variations of normality | Read a survey article on varying normality

CAUTIONARY NOTE: There is a paper where the term transitively normal is used for what we call intermediately subnormal-to-normal subgroup

Examples

Extreme examples

Every group is transitively normal as a subgroup of itself.
The trivial subgroup is transitively normal in any group.

Examples in small finite groups

Below are some examples of a proper nontrivial subgroup that satisfy the property transitively normal subgroup.

	Group part	Subgroup part	Quotient part
Center of dihedral group:D8	Dihedral group:D8	Cyclic group:Z2	Klein four-group

Below are some examples of a proper nontrivial subgroup that does not satisfy the property transitively normal subgroup.

Metaproperties

BEWARE! This section of the article uses terminology local to the wiki, possibly without giving a full explanation of the terminology used (though efforts have been made to clarify terminology as much as possible within the particular context)

Metaproperty name	Satisfied?	Proof	Statement with symbols
transitive subgroup property	Yes	transitive normality is transitive	If $H \leq K \leq G$ are groups such that $H$ is transitively normal in $K$ and $K$ is transitively normal in $G$ , then $H$ is transitively normal in $G$ .
intermediate subgroup condition	Yes	transitive normality satisfies intermediate subgroup condition	If $H \leq K \leq G$ and $H$ is transitively normal in $G$ , then $H$ is transitively normal in $K$ .
finite-intersection-closed subgroup property	No	transitive normality is not finite-intersection-closed	It is possible to have a group $G$ and transitively normal subgroups $H_{1}, H_{2}$ of $G$ such that $H_{1} \cap H_{2}$ is not transitively normal in $G$ .
finite-join-closed subgroup property	No	transitive normality is not finite-join-closed	It is possible to have a group $G$ and transitively normal subgroups $H_{1}, H_{2}$ of $G$ such that the join $⟨ H_{1}, H_{2} ⟩$ (which is the same as the product $H_{1} H_{2}$ ) is not transitively normal.
centralizer-closed subgroup property	No	transitive normality is not centralizer-closed	It is possible to have a group $G$ and a transitively normal subgroup $H$ of $G$ such that the centralizer $C_{G} (H)$ is not transitively normal.
image condition	Yes	transitive normality satisfies image condition	If $H$ is a transitively normal subgroup of $G$ and $φ : G \to K$ is a surjective homomorphism, then $φ (H)$ is a transitively normal subgroup of $K$ .
quotient-transitive subgroup property	No	transitive normality is not quotient-transitive	It is possible to have groups $H \leq K \leq G$ such that $H$ is transitively normal in $G$ and $K / H$ is transitively normal in $G / H$ , but $K$ is not transitively normal in $G$ .

Relation with other properties

Conjunction with other properties

Here are some conjunctions with other subgroup properties:

Conjunction	Other component of conjunction	Intermediate notions	Additional comments
characteristic transitively normal subgroup	characteristic subgroup	\|FULL LIST, MORE INFO
c-closed transitively normal subgroup	c-closed subgroup (equals its double centralizer)	\|FULL LIST, MORE INFO

Conjunctions with group properties:

Conjunction	Property of the group	Intermediate notions	Additional comments
cyclic normal subgroup	Cyclic group	\|FULL LIST, MORE INFO	for a cyclic subgroup, being normal is equivalent to being transitively normal.
abelian hereditarily normal subgroup	Abelian group	\|FULL LIST, MORE INFO	for an abelian subgroup, being transitively normal is equivalent to being hereditarily normal.
hereditarily normal subgroup	Dedekind group (every subgroup is normal)	\|FULL LIST, MORE INFO
nilpotent transitively normal subgroup	Nilpotent group	\|FULL LIST, MORE INFO

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
central factor	product with centralizer is whole group	central factor implies transitively normal	transitively normal not implies central factor (see also list of examples)	\|FULL LIST, MORE INFO
direct factor	factor in an internal direct product	direct factor implies transitively normal (also, via central factor)	(via central factor) (see also list of examples)	\|FULL LIST, MORE INFO
cocentral subgroup	product with center is whole group	(via central factor)	(via central factor)	\|FULL LIST, MORE INFO
central subgroup	contained in the center	(via central factor, via hereditarily normal)	(via central factor, via hereditarily normal)	\|FULL LIST, MORE INFO
locally inner autorphism-balanced subgroup	every inner automorphism of the whole group restricts to a locally inner automorphism			\|FULL LIST, MORE INFO
conjugacy-closed normal subgroup	both conjugacy-closed and normal	conjugacy-closed normal implies transitively normal	transitively normal not implies conjugacy-closed	\|FULL LIST, MORE INFO
SCAB-subgroup	any subgroup-conjugating automorphism of whole group restricts to subgroup-conjugating automorphism of subgroup	SCAB implies transitively normal	Transitively normal not implies SCAB	\|FULL LIST, MORE INFO
hereditarily normal subgroup	every subgroup of it is normal in whole group			\|FULL LIST, MORE INFO
cyclic normal subgroup	cyclic and normal	(via SCAB, via hereditarily normal)	(via SCAB, via hereditarily normal)	\|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
normal subgroup		(by definition)	normality is not transitive	\|FULL LIST, MORE INFO
CEP-subgroup	every normal subgroup of it arises as its intersection with a normal subgroup of whole group			\|FULL LIST, MORE INFO
subgroup whose commutator with any subset is normal	its commutator with any subset of whole group is normal in whole group	commutator of a transitively normal subgroup and a subset implies normal	commutator with any subset is normal not implies transitively normal	\|FULL LIST, MORE INFO

Formalisms

BEWARE! This section of the article uses terminology local to the wiki, possibly without giving a full explanation of the terminology used (though efforts have been made to clarify terminology as much as possible within the particular context)

Function restriction expression

This subgroup property is a function restriction-expressible subgroup property: it can be expressed by means of the function restriction formalism, viz there is a function restriction expression for it.
Find other function restriction-expressible subgroup properties | View the function restriction formalism chart for a graphic placement of this property

Function restriction expression	$H$ is a transitively normal subgroup of $G$ if ...	This means that transitive normality is ...	Additional comments
normal automorphism $\to$ normal automorphism	every normal automorphism of $G$ restricts to a normal automorphism of $H$	the balanced subgroup property for normal automorphisms	Hence, it is a t.i. subgroup property, both transitive and identity-true
inner automorphism $\to$ normal automorphism	every inner automorphism of $G$ restricts to a normal automorphism of $H$
class-preserving automorphism $\to$ normal automorphism	every class-preserving automorphism of $G$ restricts to a normal automorphism of $H$
subgroup-conjugating automorphism $\to$ normal automorphism	every subgroup-conjugating automorphism of $G$ restricts to a normal automorphism of $H$
strong monomial automorphism $\to$ normal automorphism	every strong monomial automorphism of $G$ restricts to a normal automorphism of $H$

@@ Line 8: / Line 8: @@
 ! No. !! Shorthand !! A [[subgroup]] of a [[group]] is termed transitively normal if ... !! A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed transitively normal if ...
 |-
-| 1 || transitively normal || every [[defining ingredient::normal subgroup]] of the subgroup is normal in the whole group || whenever <math>K</math> is a [[normal subgroup]] of <math>H</math>, <math>K</math> is also a normal subgroup of <math>G</math>.
+| 1 || transitively normal || every [[defining ingredient::normal subgroup]] of the subgroup is normal in the whole group. || whenever <math>K</math> is a [[normal subgroup]] of <math>H</math>, <math>K</math> is also a normal subgroup of <math>G</math>.
 |-
 | 2 || normal automorphisms restriction || every  [[defining ingredient::normal automorphism]] of the whole group restricts to a normal automorphism of the subgroup. || for any [[normal automorphism]] <math>\sigma</math> of <math>G</math>, the restriction of <math>\sigma</math> to <math>H</math> is a normal automorphism of <math>H</math>.
@@ Line 43: / Line 43: @@
 | [[satisfies metaproperty::transitive subgroup property]] || Yes || [[transitive normality is transitive]] || If <math>H \le K \le G</math> are groups such that <math>H</math> is transitively normal in <math>K</math> and <math>K</math> is transitively normal in <math>G</math>, then <math>H</math> is transitively normal in <math>G</math>.
 |-
-| [[satisfies metaproperty::intermediate subgroup condition]] || Yes || [[transitive normality satisfies intermediate subgroup condition]] || If <math>H \le K \le G</math> and <math>H</math> is transitively normal in <math>G</math>, then <math>H</math> is tansitively normal in <math>K</math>.
+| [[satisfies metaproperty::intermediate subgroup condition]] || Yes || [[transitive normality satisfies intermediate subgroup condition]] || If <math>H \le K \le G</math> and <math>H</math> is transitively normal in <math>G</math>, then <math>H</math> is transitively normal in <math>K</math>.
 |-
 | [[dissatisfies metaproperty::finite-intersection-closed subgroup property]] || No || [[transitive normality is not finite-intersection-closed]] || It is possible to have a group <math>G</math> and transitively normal subgroups <math>H_1,H_2</math> of <math>G</math> such that <math>H_1 \cap H_2</math> is not transitively normal in <math>G</math>.
@@ Line 51: / Line 51: @@
 | [[dissatisfies metaproperty::centralizer-closed subgroup property]] || No || [[transitive normality is not centralizer-closed]] || It is possible to have a group <math>G</math> and a transitively normal subgroup <math>H</math> of <math>G</math> such that the [[centralizer]] <math>C_G(H)</math> is not transitively normal.
 |-
-| [[satisfies metaproperty::image condition]] || Yes || [[transitive normalizer satisfies image condition]] || If <math>H</math> is a transitively normal subgroup of <math>G</math> and <math>\varphi:G \to K</math> is a surjective homomorphism, then <math>\varphi(H)</math> is a transitively normal subgroup of <math>K</math>.
+| [[satisfies metaproperty::image condition]] || Yes || [[transitive normality satisfies image condition]] || If <math>H</math> is a transitively normal subgroup of <math>G</math> and <math>\varphi:G \to K</math> is a surjective homomorphism, then <math>\varphi(H)</math> is a transitively normal subgroup of <math>K</math>.
 |-
 | [[dissatisfies metaproperty::quotient-transitive subgroup property]] || No || [[transitive normality is not quotient-transitive]] || It is possible to have groups <math>H \le K \le G</math> such that <math>H</math> is transitively normal in <math>G</math> and <math>K/H</math> is transitively normal in <math>G/H</math>, but <math>K</math> is not transitively normal in <math>G</math>.
 |}
 ==Relation with other properties==
@@ Line 95: / Line 96: @@
 |-
 | [[Weaker than::central subgroup]] || contained in the [[center]] || (via central factor, via hereditarily normal) || (via central factor, via hereditarily normal) || {{intermediate notions short|transitively normal subgroup|central subgroup}}
+|-
+| [[Weaker than::locally inner autorphism-balanced subgroup]] || every [[inner automorphism]] of the whole group restricts to a [[locally inner automorphism]] || || || {{intermediate notions short|transitively normal subgroup|locally inner automorphism-balanced subgroup}}
 |-
 | [[Weaker than::conjugacy-closed normal subgroup]] ||  both [[conjugacy-closed subgroup|conjugacy-closed]] and [[normal subgroup|normal]] || [[conjugacy-closed normal implies transitively normal]] || [[transitively normal not implies conjugacy-closed]] || {{intermediate notions short|transitively normal subgroup|conjugacy-closed normal subgroup}}