Characteristic implies normal: Difference between revisions

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This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., characteristic subgroup) must also satisfy the second subgroup property (i.e., normal subgroup)
View all subgroup property implications | View all subgroup property non-implications
Get more facts about characteristic subgroup|Get more facts about normal subgroup

To learn more about the similarities and differences between characteristicity and normality, refer the survey article Characteristic versus normal

Statement

Let $H$ be a characteristic subgroup of $G$ . Then, $H$ is normal in $G$ .

Definitions used

Characteristic subgroup

Further information: Characteristic subgroup

The definitions we use here are as follows:

Hands-on definition: A subgroup $H$ of a group $G$ is termed a characteristic subgroup, if for any automorphism $\sigma$ of $G$ , we have $\sigma (H)=H$ .
Definition using function restriction expression: We can write characteristicity as the invariance property with respect to automorphisms:

Characteristic = Automorphism $\to$ Function

This is interpreted as: any automorphism from the whole group to itself, restricts to a function from the subgroup to itself. In other words, the subgroup is invariant under automorphisms.

Definition using relation implication expression: We can write characteristicity as:

Automorphic subgroups $\implies$ Equal

In other words, any subgroup obtained by taking the image of this subgroup under an automorphism of the whole group, must be equal to it.

Definition in terms of equivalence classes of elements: A subgroup is characteristic if and only if it is the union of equivalence classes of elements under the action of the automorphism group.

Normal subgroup

Further information: Normal subgroup

The definitions we use here are as follows:

Hands-on definition: A subgroup $H$ of a group $G$ is termed normal, if for any $g\in G$ , the inner automorphism $c_{g}$ defined by conjugation by $g$ , namely the map $x\mapsto gxg^{-1}$ , gives an isomorphism on $H$ . In other words, for any $g\in G$ :

$c_{g}(H)=H$

or more explicitly:

$gHg^{-1}=H$

Implicit in this definition is the fact that $c_{g}$ is an automorphism. Further information: Group acts as automorphisms by conjugation

Definition using function restriction expression: We can write normality as the invariance property with respect to inner automorphisms:

Normal = Inner automorphism $\to$ Function

In other words, any inner automorphism on the whole group restricts to a function from the subgroup to itself.

Definition using relation implication expression: We can write:

Normal = Conjugate subgroups $\implies$ Equal

In other words, any subgroup conjugate to the given one, must be equal to it.

Definition using equivalence classes of elements: A subgroup is normal if and only if it is a union of conjugacy classes of elements.

Related facts

Related facts about groups

Characteristic of normal implies normal: A characteristic subgroup of a normal subgroup is normal in the whole group.
Left transiter of normal is characteristic: In fact, characteristicity is precisely the property needed to be a left transiter for normality. Explicitly, if $H\leq K$ is a subgroup such that whenever $K$ is normal in a group $G$ , so is $H$ , then $H$ mustbe characteristic in $G$ .
Normal not implies characteristic: A normal subgroup of a group need not be characteristic in the group.

Analogues in other algebraic structures

Derivation-invariant implies ideal is the analogue for Lie rings.

Facts used

Group acts as automorphisms by conjugation: This states that every inner automorphism of a group is an automorphism.

Proof

Hands-on proof

Given: $H$ is a characteristic subgroup of $G$ . In other words, for any automorphism $\sigma$ of $G$ , $\sigma (H)=H$ .

To prove: For any $g\in G$ , $gHg^{-1}=H$ . In other words, if $c_{g}$ denotes conjugation by $g$ i.e. the map $x\mapsto gxg^{-1}$ , then $c_{g}(H)=H$

Proof: $c_{g}$ is an inner automorphism, so it is an automorphism. Thus, invoking characteristicity, we have $c_{g}(H)=H$ , i.e. $gHg^{-1}=H$ .

Thus $H$ is a normal subgroup of $G$ .

Using function restriction expressions

This subgroup property implication can be proved by using function restriction expressions for the subgroup properties
View other implications proved this way |read a survey article on the topic

Normality is the invariance property with respect to inner automorphisms, and characteristicity is the invariance property with respect to automorphisms. Explicitly:

Normal = Inner automorphism $\to$ Function

Characteristic = Automorphism $\to$ Function

Since the left side of normality implies the left side of characteristicity, every characteristic subgroup is normal.

Using relation implication expressions

This subgroup property implication can be proved using relation implication expressions for the subgroup properties
View other implications proved in this way OR Read a survey article on the topic

The relation implication expression for normality is:

Conjugate subgroups $\implies$ Equal

The relation implication expression for characteristicity is:

Automorphic subgroups $\implies$ Equal

Since the left side for the expression for normality is stronger than the left side for the expression for characteristicity, and the right sides are the same, the subgroup property of being characteristic implies the subgroup property of being normal.

In terms of equivalence classes of elements

A normal subgroup is a subgroup that is a union of conjugacy classes; a characteristic subgroup is a subgroup that is a union of automorphism classes. Since every automorphism class is a union of conjugacy classes, every characteristic subgroup is normal.

Related properties

Normal-to-characteristic

Further information: Subnormal-to-normal and normal-to-characteristic

A normal-to-characteristic subgroup is a subgroup that, if normal, is also characteristic. An intermediately normal-to-characteristic subgroup is a subgroup that, if normal in any intermediate subgroup, is also characteristic in that intermediate subgroup. There are a number of subgroup properties that are stronger than the property of being normal-to-characteristic: in other words, any normal subgroup satisfying the property is also characteristic.

These include, for instance, the property of being automorph-conjugate, procharacteristic, paracharacteristic, core-characteristic, closure-characteristic, and many others. For more information on such properties, refer subnormal-to-normal and normal-to-characteristic.

Intermediate invariance properties

The invariance property for an automorphism property that is weaker than the property of being an inner automorphism, lies somewhere between characteristicity and normality. Here are some examples:

Monomial automorphism-invariant subgroup is a subgroup that is invariant under every monomial automorphism. It turns out that monomial automorphism-invariant subgroups are the same as normal subgroups.
Normal-extensible automorphism-invariant subgroup is a subgroup that is invariant under every normal-extensible automorphism.
Characteristic-extensible automorphism-invariant subgroup is a subgroup that is invariant under every characteristic-extensible automorphism.
IA-automorphism-invariant subgroup is a subgroup that is invariant under all IA-automorphisms.
Cofactorial automorphism-invariant subgroup is a subgroup that is invariant under all cofactorial automorphisms.

Properties involving additional structure

Invariance properties with respect to automorphisms that preserve additional structure imposed on the group, lie between characteristicity and normality. For instance:

Topologically characteristic subgroup in a topological group

Other intermediate properties

Potentially characteristic equals normal: A potentially characteristic subgroup is a subgroup that is characteristic in some bigger group containing the ambient group. A subgroup of a group is potentially characteristic iff it is normal.
Characteristic-potentially characteristic subgroup is a subgroup such that there is a group containing the bigger group in which both of them are characteristic.
Normal-potentially characteristic subgroup is a subgroup such that there is a group containing the bigger group as a normal subgroup and the smaller group as a characteristic subgroup.

References

Textbook references

Groups and representations by Jonathan Lazare Alperin and Rowen B. Bell, ISBN 0387945261, ^{More info}, Page 17
Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347, ^{More info}, Page 135, Page 137 (Problem 6)
Topics in Algebra by I. N. Herstein, ^{More info}, Page 70, Problem 7(a)
A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613, ^{More info}, Page 28, 1.5.6(i)
Algebra by Michael Artin, ISBN 0130047635, 13-digit ISBN 978-0130047632, ^{More info}, Page 234, Exercise 7 of Section 8 (Generators and relations)
Nilpotent groups and their automorphisms by Evgenii I. Khukhro, ISBN 3110136724, ^{More info}, Page 4, Section 1.1 (statement and proof as passing mention in paragraph)

@@ Line 1: / Line 1: @@
 {{trivial result}}
-{{subgroup property implication}}
+{{subgroup property implication|
+stronger = characteristic subgroup|
+weaker = normal subgroup|
+stronger relevance = 1|
+weaker relevance = 1}}
+[[difficulty level::1| ]]
 {{quotation|''To learn more about the similarities and differences between characteristicity and normality, refer the survey article [[Characteristic versus normal]]''}}
 ==Statement==
-===Verbal statement===
+Let <math>H</math> be a [[characteristic subgroup]] of <math>G</math>. Then, <math>H</math> is [[normal subgroup|normal]] in <math>G</math>.
-Every [[characteristic subgroup]] of a group is a [[normal subgroup]].
-===Symbolic statement===
-Let <math>H</math> be a characteristic subgroup of <math>G</math>. Then, <math>H</math> is [[normal subgroup|normal]] in <math>G</math>.
-===Property-theoretic statement===
-The [[subgroup property]] of being [[characteristic subgroup|characteristic]] is ''stronger'' than the subgroup property of being [[normal subgroup|normal]].
 ==Definitions used==
@@ Line 25: / Line 20: @@
 The definitions we use here are as follows:
-* ''Hands-on definition'': A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed a characteristic subgroup, if for any [[automorphism]] <math>\sigma</math> of <math>G</math>, we have <math>\sigma(H) = H</math>.
+* '''Hands-on definition''': A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed a characteristic subgroup, if for any [[automorphism]] <math>\sigma</math> of <math>G</math>, we have <math>\sigma(H) = H</math>.
-* ''Definition using [[function restriction expression]]'': We can write characteristicity as the [[invariance property]] with respect to automorphisms:
+* '''Definition using [[function restriction expression]]''': We can write characteristicity as the [[invariance property]] with respect to automorphisms:
 Characteristic = Automorphism <math>\to</math> Function
 This is interpreted as: any automorphism from the whole group to itself, ''restricts'' to a function from the subgroup to itself. In other words, the subgroup is invariant under automorphisms.
-* ''Definition using [[relation implication expression]]'': We can write characteristicity as:
+* '''Definition using [[relation implication expression]]''': We can write characteristicity as:
 [[Automorphic subgroups]] <math>\implies</math> Equal
 In other words, any subgroup obtained by taking the image of this subgroup under an automorphism of the whole group, must be equal to it.
+* '''Definition in terms of equivalence classes of elements''': A subgroup is characteristic if and only if it is the union of equivalence classes of elements under the action of the automorphism group.
-===Definitions for normal===
+===Normal subgroup===
 {{further|[[Normal subgroup]]}}
@@ Line 43: / Line 39: @@
 The definitions we use here are as follows:
-* ''Hands-on definition'': A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed normal, if for any <math>g \in G</math>, the [[inner automorphism]] <math>c_g</math> defined by conjugation by <math>g</math>, namely the map <math>x \mapsto gxg^{-1}</math>, gives an isomorphism on <math>H</math>. In other words, for any <math>g \in G</math>:
+* '''Hands-on definition''': A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed normal, if for any <math>g \in G</math>, the [[inner automorphism]] <math>c_g</math> defined by conjugation by <math>g</math>, namely the map <math>x \mapsto gxg^{-1}</math>, gives an isomorphism on <math>H</math>. In other words, for any <math>g \in G</math>:
 <math>c_g(H) = H</math>
@@ Line 51: / Line 47: @@
 <math>gHg^{-1} = H</math>
-Implicit in this definition is the fact that <math>c_g</math> ''is'' an automorphism. {{fillin}}
+Implicit in this definition is the fact that <math>c_g</math> ''is'' an automorphism. {{further|[[Group acts as automorphisms by conjugation]]}}
-* ''Definition using [[function restriction expression]]'': We can write normality as the invariance property with respect to inner automorphisms:
+* '''Definition using [[function restriction expression]]''': We can write normality as the invariance property with respect to inner automorphisms:
 Normal = [[Inner automorphism]] <math>\to</math> Function
@@ Line 59: / Line 55: @@
 In other words, any inner automorphism on the whole group ''restricts'' to a function from the subgroup to itself.
-* ''Definition using [[relation implication expression]]'': We can write:
+* '''Definition using [[relation implication expression]]''': We can write:
 Normal = [[Conjugate subgroups]] <math>\implies</math> Equal
 In other words, any subgroup conjugate to the given one, must be equal to it.
+* '''Definition using equivalence classes of elements''': A subgroup is normal if and only if it is a union of [[conjugacy class]]es of elements.
+==Related facts==
+===Related facts about groups===
+* [[Characteristic of normal implies normal]]: A characteristic subgroup ''of'' a normal subgroup is normal in the whole group.
+* [[Left transiter of normal is characteristic]]: In fact, characteristicity is ''precisely'' the property needed to be a [[left transiter]] for normality. Explicitly, if <math>H \le K</math> is a subgroup such that whenever <math>K</math> is normal in a group <math>G</math>, so is <math>H</math>, then <math>H</math> mustbe characteristic in <math>G</math>.
+* [[Normal not implies characteristic]]: A normal subgroup of a group need not be characteristic in the group.
+===Analogues in other algebraic structures===
+* [[Derivation-invariant implies ideal]] is the analogue for [[Lie ring]]s.
+==Facts used==
+# [[uses::Group acts as automorphisms by conjugation]]: This states that every inner automorphism of a group ''is'' an automorphism.
 ==Proof==
@@ Line 70: / Line 82: @@
 ===Hands-on proof===
-''Given'': Let <math>H</math> be a characteristic subgroup of <math>G</math>. Then, for any automorphism <math>\sigma</math> of <math>G</math>, <math>\sigma(H) = H</math>. Our goal is to show that for any <math>g \in G</math>, <math>gHg^{-1} = H</math>.
+'''Given''': <math>H</math> is a characteristic subgroup of <math>G</math>. In other words, for any automorphism <math>\sigma</math> of <math>G</math>, <math>\sigma(H) = H</math>.
-''To prove'': For any <math>g \in G</math>, for <math>c_g</math> denotes conjugation by <math>g</math>  i.e. the map <math>x \mapsto gxg^{-1}</math>.
+'''To prove''': For any <math>g \in G</math>, <math>gHg^{-1} = H</math>. In other words, if <math>c_g</math> denotes conjugation by <math>g</math>  i.e. the map <math>x \mapsto gxg^{-1}</math>, then <math>c_g(H) = H</math>
-''Proof'': <math>c_g</math> is an [[inner automorphism]], so it is an automorphism. Thus, invoking characteristicity, we have <math>c_g(H) = H</math>, i.e. <math>gHg^{-1} =H</math>.
+'''Proof''': <math>c_g</math> is an [[inner automorphism]], so it is an automorphism. Thus, invoking characteristicity, we have <math>c_g(H) = H</math>, i.e. <math>gHg^{-1} =H</math>.
 Thus <math>H</math> is a normal subgroup of <math>G</math>.
@@ Line 90: / Line 102: @@
 {{riexp implication}}
-A subgroup is normal if any subgroup that ''resembles'' it as a conjugate is the same as it; a subgroup is characteristic if any subgroup that ''resembles'' it via an automorphism is the same as it. Since the notion of resembling as a conjugate is stronger than the notion of resembling via an automorphism, every characteristic subgroup is normal.
+The relation implication expression for normality is:
+[[Conjugate subgroups]] <math>\implies</math> Equal
+The relation implication expression for characteristicity is:
+[[Automorphic subgroups]] <math>\implies</math> Equal
+Since the left side for the expression for normality is ''stronger than'' the left side for the expression for characteristicity, and the right sides are the same, the subgroup property of being characteristic implies the subgroup property of being normal.
 ===In terms of equivalence classes of elements===
@@ Line 96: / Line 116: @@
 A normal subgroup is a subgroup that is a union of conjugacy classes; a characteristic subgroup is a subgroup that is a union of automorphism classes. Since every automorphism class is a union of conjugacy classes, every characteristic subgroup is normal.
-==Converse==
+==Related properties==
-===Normal subgroups need not be characteristic===
-{{further|[[Normal not implies characteristic]]}}
 ===Normal-to-characteristic===
-A property that, along with normality, implies characterisicity, is termed a [[normal-to-characteristic subgroup property]]. A typical example is the property of being [[automorph-conjugate subgroup|automorph-conjugate]]. This property plays on the ''resemblance''-based definitions of normality and characteristicity.
+{{further|[[Subnormal-to-normal and normal-to-characteristic]]}}
-===Groups where every normal subgroup is characteristic===
+A [[normal-to-characteristic subgroup]] is a subgroup that, if normal, is also characteristic. An [[intermediately normal-to-characteristic subgroup]] is a subgroup that, if normal in any intermediate subgroup, is also characteristic in that intermediate subgroup. There are a number of subgroup properties that are stronger than the property of being normal-to-characteristic: in other words, any normal subgroup satisfying the property is also characteristic.
-A group in which the normal subgroups are precisely the same as the characteristic subgroups, is termed a [[N=C-group]].
+These include, for instance, the property of being [[automorph-conjugate subgroup|automorph-conjugate]], [[procharacteristic subgroup|procharacteristic]], [[paracharacteristic subgroup|paracharacteristic]], [[core-characteristic subgroup|core-characteristic]], [[closure-characteristic subgroup|closure-characteristic]], and many others. For more information on such properties, refer [[subnormal-to-normal and normal-to-characteristic]].
-==Intermediate properties==
 ===Intermediate invariance properties===
@@ Line 116: / Line 130: @@
 The invariance property for an [[automorphism property]] that is weaker than the property of being an inner automorphism, lies somewhere between characteristicity and normality. Here are some examples:
-* [[Extensible automorphism-invariant subgroup]] is a subgroup that is invariant under every [[extensible automorphism]] (a property of automorphisms that is weaker than the property of being an [[inner automorphism]])
+* '''Monomial automorphism-invariant subgroup''' is a subgroup that is invariant under every [[monomial automorphism]]. It turns out that monomial automorphism-invariant subgroups are the same as normal subgroups.
-* [[Monomial automorphism-invariant subgroup]] is a subgroup that is invariant under every [[monomial automorphism]]
+* [[Normal-extensible automorphism-invariant subgroup]] is a subgroup that is invariant under every [[normal-extensible automorphism]].
+* [[Characteristic-extensible automorphism-invariant subgroup]] is a subgroup that is invariant under every [[characteristic-extensible automorphism]].
+* [[IA-automorphism-invariant subgroup]] is a subgroup that is invariant under all [[IA-automorphism]]s.
+* [[Cofactorial automorphism-invariant subgroup]] is a subgroup that is invariant under all [[cofactorial automorphism]]s.
 ===Properties involving additional structure===
@@ Line 127: / Line 144: @@
 ===Other intermediate properties===
-* [[Potentially characteristic subgroup]] is a subgroup that is characteristic in some group containing the bigger group.
+* [[Potentially characteristic equals normal]]: A ''potentially'' characteristic subgroup is a subgroup that is characteristic in some bigger group containing the ambient group. A subgroup of a group is potentially characteristic iff it is normal.
-* [[Strongly potentially characteristic subgroup]] is a subgroup such that there is a group containing the bigger group in which both of them are characteristic.
+* [[Characteristic-potentially characteristic subgroup]] is a subgroup such that there is a group containing the bigger group in which both of them are characteristic.
+* [[Normal-potentially characteristic subgroup]] is a subgroup such that there is a group containing the bigger group as a normal subgroup and the smaller group as a characteristic subgroup.
 ==References==
 ===Textbook references===
-* {{booklink|AlperinBell}}, Page 17
+* {{booklink-stated|AlperinBell}}, Page 17
-* {{booklink|DummitFoote}}, Page 135, Page 137 (Problem 6)
+* {{booklink-stated|DummitFoote}}, Page 135, Page 137 (Problem 6)
-* {{booklink|Herstein}}, Page 70, Problem 7(a)
+* {{booklink-stated|Herstein}}, Page 70, Problem 7(a)
+* {{booklink-proved|RobinsonGT}}, Page 28, 1.5.6(i)
+* {{booklink-stated|Artin}}, Page 234, Exercise 7 of Section 8 (''Generators and relations'')
+* {{booklink-proved|KhukhroNGA}}, Page 4, Section 1.1 (statement and proof as passing mention in paragraph)