Hereditarily normal subgroup: Difference between revisions

Latest revision as of 17:15, 18 January 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 article describes a property that arises as the conjunction of a subgroup property: transitively normal subgroup with a group property (itself viewed as a subgroup property): Dedekind group
View a complete list of such conjunctions

Definition

Symbol-free definition

A subgroup of a group is termed hereditarily normal or quasicentral if it satisfies the following equivalent conditions:

Every subgroup of the subgroup is a normal subgroup of the whole group.
It is both a transitively normal subgroup of the whole group and a Dedekind group.
All inner automorphisms of the whole group restrict to power automorphisms on the subgroup.

Definition with symbols

A subgroup $H$ of a group $G$ is termed hereditarily normal or quasicentral if it satisfies the following equivalent conditions:

For every subgroup $K$ of $H$ , $K$ is a normal subgroup of $G$ .
$H$ is both a Dedekind group and a transitively normal subgroup of $G$ .
For any $g\in G$ , the inner automorphism of $G$ given by conjugation by $g$ restricts to a power automorphism of $H$ .

Formalisms

In terms of the hereditarily operator

This property is obtained by applying the hereditarily operator to the property: normal subgroup
View other properties obtained by applying the hereditarily operator

The property of being hereditarily normal is a result of applying the hereditarily operator on the property of normality.

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 hereditarily normal subgroup of $G$ if ...	This means that "hereditarily normal" is ...	Additional comments
inner automorphism $\to$ power automorphism	every inner automorphism of $G$ restricts to a power automorphism of $H$
class-preserving automorphism $\to$ power automorphism	every class-preserving automorphism of $G$ restricts to a power automorphism of $H$
subgroup-conjugating automorphism $\to$ power automorphism	every subgroup-conjugating automorphism of $G$ restricts to a power automorphism of $H$
normal automorphism $\to$ power automorphism	every normal automorphism of $G$ restricts to a power automorphism of $H$

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Central subgroup	contained in the center	central implies hereditarily normal	hereditarily normal not implies central	\|FULL LIST, MORE INFO
Cyclic normal subgroup	normal and cyclic as a group			\|FULL LIST, MORE INFO
Abelian hereditarily normal subgroup	abelian subgroup and every subgroup of it is normal in whole group			\|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
SCAB-subgroup	every subgroup-conjugating automorphism of the whole group restricts to a subgroup-conjugating automorphism of the subgroup	hereditarily normal implies SCAB	SCAB not implies hereditarily normal	\|FULL LIST, MORE INFO
Transitively normal subgroup	every normal subgroup of it is normal in the whole group			\|FULL LIST, MORE INFO
Hereditarily permutable subgroup	every subgroup of it is permutable in the whole group			\|FULL LIST, MORE INFO
Hereditarily subnormal subgroup	every subgroup of it is subnormal in the whole group			\|FULL LIST, MORE INFO
Hereditarily pronormal subgroup	every subgroup of it is pronormal in the whole group			\|FULL LIST, MORE INFO
Normal subgroup				\|FULL LIST, MORE INFO

Metaproperties

Left-hereditariness

This subgroup property is left-hereditary: any subgroup of a subgroup with this property also has this property. Hence, it is also a transitive subgroup property.

Since the left-hereditarily operator is idempotent, the property of being hereditarily normal is itself left hereditary (that is, every subgroup of a hereditarily normal subgroup is hereditarily normal).

Note that being a left-hereditary property, it is automatically transitive and also intersection-closed.

Transfer condition

YES: This subgroup property satisfies the transfer condition: if a subgroup has the property in the whole group, its intersection with any subgroup has the property in that subgroup.
View other subgroup properties satisfying the transfer condition

Since normality satisfies the intermediate subgroup condition, and the left-hereditarily operator preserves the intermediate subgroup condition, the property of being hereditarily normal also satisfies the transfer. condition. Hence it also satisfies the intermediate subgroup condition.

Trimness

The trivial group is obviously hereditarily normal.

The whole group is hereditarily normal if and only if every subgroup of the group is normal. Groups with this property are either Abelian groups or Hamiltonian groups.

@@ Line 1: / Line 1: @@
+{{semistddef}}
 {{subgroup property}}
+{{group-subgroup property conjunction|transitively normal subgroup|Dedekind group}}
 ==Definition==
 ===Symbol-free definition===
-A subgroup of a group is termed '''hereditarily normal''' (sometimes, '''quasicentral''') if every subgroup of it is normal in the whole group.
+A subgroup of a group is termed '''hereditarily normal''' or '''quasicentral''' if it satisfies the following equivalent conditions:
+# Every subgroup of the subgroup is a [[defining ingredient::normal subgroup]] of the whole group.
+# It is both a [[transitively normal subgroup]] of the whole group and a [[Dedekind group]].
+# All [[inner automorphism]]s of the whole group restrict to [[power automorphism]]s on the subgroup.
 ===Definition with symbols===
-A subgroup <math>H</math> of a group <math>G</math> is termed '''hereditarily normal''' (sometimes, '''quasicentral''') if for any subgroup <math>K \le H</math>, <math>K</math> is normal in <math>G</math>.
+A subgroup <math>H</math> of a group <math>G</math> is termed '''hereditarily normal''' or '''quasicentral''' if it satisfies the following equivalent conditions:
+# For every subgroup <math>K</math> of <math>H</math>, <math>K</math> is a [[normal subgroup]] of <math>G</math>.
+# <math>H</math> is both a [[Dedekind group]] and a [[transitively normal subgroup]] of <math>G</math>.
+# For any <math>g \in G</math>, the [[inner automorphism]] of <math>G</math> given by conjugation by <math>g</math> restricts to a [[power automorphism]] of <math>H</math>.
 ==Formalisms==
@@ Line 15: / Line 24: @@
 The property of being hereditarily normal is a result of applying the [[hereditarily operator]] on the property of normality.
+{{frexp}}
+{| class="wikitable" border="1"
+! Function restriction expression !! <math>H</math> is a hereditarily normal subgroup of <math>G</math> if ... !! This means that "hereditarily normal" is ... !! Additional comments
+|-
+| [[left side of function restriction expression::inner automorphism]] <math>\to</math> [[right side of function restriction expression::power automorphism]] || every inner automorphism of <math>G</math> restricts to a power automorphism of <math>H</math> || ||
+|-
+| [[left side of function restriction expression::class-preserving automorphism]] <math>\to</math> [[right side of function restriction expression::power automorphism]] || every class-preserving automorphism of <math>G</math> restricts to a power automorphism of <math>H</math> || ||
+|-
+| [[left side of function restriction expression::subgroup-conjugating automorphism]] <math>\to</math> [[right side of function restriction expression::power automorphism]] || every subgroup-conjugating automorphism of <math>G</math> restricts to a power automorphism of <math>H</math> || ||
+|-
+| [[left side of function restriction expression::normal automorphism]] <math>\to</math> [[right side of function restriction expression::power automorphism]] || every normal automorphism of <math>G</math> restricts to a power automorphism of <math>H</math> || ||
+|}
 ==Relation with other properties==
@@ Line 20: / Line 43: @@
 ===Stronger properties===
-* [[Central subgroup]] (subgroup lying inside the [[center]])
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Weaker than::Central subgroup]] || contained in the [[center]] || [[central implies hereditarily normal]] || [[hereditarily normal not implies central]] || {{intermediate notions short|hereditarily normal subgroup|central subgroup}}
+|-
+| [[Weaker than::Cyclic normal subgroup]] || normal and [[cyclic group|cyclic as a group]] || || || {{intermediate notions short|hereditarily normal subgroup|cyclic normal subgroup}}
+|-
+| [[Weaker than::Abelian hereditarily normal subgroup]] || abelian subgroup and every subgroup of it is normal in whole group || || || {{intermediate notions short|hereditarily normal subgroup|abelian hereditarily normal subgroup}}
+|}
 ===Weaker properties===
-* [[Transitively normal subgroup]]
+{| class="sortable" border="1"
-* [[Hereditarily permutable subgroup]]
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
-* [[Normal subgroup]]
+|-
+| [[Stronger than::SCAB-subgroup]] || every [[subgroup-conjugating automorphism]] of the whole group restricts to a subgroup-conjugating automorphism of the subgroup || [[hereditarily normal implies SCAB]] || [[SCAB not implies hereditarily normal]] || {{intermediate notions short|SCAB-subgroup|hereditarily normal subgroup}}
+|-
+| [[Stronger than::Transitively normal subgroup]] || every normal subgroup of it is normal in the whole group || || || {{intermediate notions short|transitively normal subgroup|hereditarily normal subgroup}}
+|-
+| [[Stronger than::Hereditarily permutable subgroup]] || every subgroup of it is [[permutable subgroup|permutable]] in the whole group || || || {{intermediate notions short|hereditarily permutable subgroup|hereditarily normal subgroup}}
+|-
+| [[Stronger than::Hereditarily subnormal subgroup]] || every subgroup of it is [[subnormal subgroup|subnormal]] in the whole group || || || {{intermediate notions short|hereditarily subnormal subgroup|hereditarily normal subgroup}}
+|-
+| [[Stronger than::Hereditarily pronormal subgroup]] || every subgroup of it is [[pronormal subgroup|pronormal]] in the whole group || || || {{intermediate notions short|hereditarily pronormal subgroup|hereditarily normal subgroup}}
+|-
+| [[Stronger than::Normal subgroup]] || || || || {{intermediate notions short|normal subgroup|hereditarily normal subgroup}}
+|}
 ==Metaproperties==