Characteristic of normal implies normal: Difference between revisions

Latest revision as of 02:20, 6 January 2026

This article describes a computation relating the result of the Composition operator (?) on two known subgroup properties (i.e., Characteristic subgroup (?) and Normal subgroup (?)), to another known subgroup property (i.e., Normal subgroup (?))
View a complete list of composition computations

Statement

Property-theoretic statement

Characteristic * Normal $\leq$ Normal

Here, $*$ denotes the composition operator.

Verbal statement

Every characteristic subgroup of a normal subgroup is normal.

Statement with symbols

Let $H \leq K \leq G$ be groups such that $H$ is characteristic in $K$ and $K$ is normal in $G$ . Then, $H$ is normal in $G$ .

Related facts

Basic ideas implicit in the definitions

Restriction of automorphism to subgroup invariant under it and its inverse is automorphism: If $K \leq G$ is a subgroup and $σ$ is an automorphism of $G$ such that both $σ$ and $σ^{- 1}$ send $K$ to within itself, then $σ$ restricts to an automorphism of $K$ . This is the key idea used in arguing that an inner automorphism of the biggest group must restrict to an automorphism of the intermediate subgroup, rather than merely to a homomorphism from the intermediate subgroup to itself. Note that this idea is implicit in the equivalence between different formulations of the notion of normal subgroup.

Related facts in group theory

Characteristicity is transitive: A characteristic subgroup of a characteristic subgroup is characteristic.
Left transiter of normal is characteristic: Characteristicity is the weakest, or most general property, for which the above statement is true. This is made precise in the statement that characteristicity is the left transiter for normality.
Automorph-permutable of normal implies conjugate-permutable: This statement has many corollaries; for instance, 2-subnormal implies conjugate-permutable

Analogues

Derivation-invariant subring of ideal implies ideal: This is the analogous statement for Lie rings. Here, derivations play the role of automorphisms, Lie subrings play the role of subgroups, ideals play the role of normal subgroups, inner derivations play the role of inner automorphisms, and derivation-invariant subrings play the role of characteristic subgroups.

Applications

For a complete list of applications, refer:

Category:Applications of characteristic of normal implies normal

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 $σ$ of $G$ , we have $σ (H) = H$ .
Definition using function restriction expression: We can write characteristicity as the balanced subgroup property with respect to automorphisms:

Characteristic = Automorphism $\to$ Automorphism

This is interpreted as: any automorphism from the whole group to itself, restricts to an automorphism from the subgroup to itself. Note that this is stronger than simply saying that it maps the subgroup to within itself -- we also demand that the restriction be an automorphism of the subgroup.

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 $x \mapsto g x g^{- 1}$ , gives a map from $H$ to itself. In other words, for any $g \in G$ :

$c_{g} (H) \leq H$

or more explicitly:

$g H g^{- 1} \leq H$

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

Note that it turns out that the above also implies that $c_{g} (H) = H$ (This is because we have $c_{g} (H) \leq H$ as well as $c_{g^{- 1}} (H) \leq H$ ). This equivalence of ideas is crucial to the proof.

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

Normal = Inner automorphism $\to$ Automorphism

In other words, any inner automorphism on the whole group restricts to an automorphism from the subgroup to itself. Note that this is stronger than saying that the inner automorphism simply sends the subgroup to itself -- we also demand that the restriction itself be an automorphism of the subgroup.

Facts used

Restriction of automorphism to subgroup invariant under it and its inverse is automorphism
Composition rule for function restriction: This is used for the proof using function restriction expressions.

Proof

This proof uses a tabular format for presentation. Provide feedback on tabular proof formats in a survey (opens in new window/tab) | Learn more about tabular proof formats|View all pages on facts with proofs in tabular format

Hands-on proof

Given: Groups $H \leq K \leq G$ such that $H$ is characteristic in $K$ and $K$ is normal in $G$ . An element $g \in G$ .

To Prove: The map $g \in G$ , the map $c_{g} : x \mapsto g x g^{- 1}$ maps $H$ to $H$ (and in fact, yields an automorphism of $H$ ).

Proof:

Step no.	Assertion/construction	Facts used	Given data used	Previous steps used	Explanation
1	$c_{g} (x) \in K$ for every $x \in K$ and $c_{g}$ restricts to an automorphism of $K$ . Call this automorphism $σ$ .	definition of normal subgroup Fact (1)	$K$ is normal in $G$ $g$ is in $G$ .		[SHOW MORE] By the definition of normal, $c_{g}$ sends $K$ to $K$ . So does $c_{g^{- 1}}$ . The two conjugation maps are inverses of each other, so by Fact (1), $c_{g}$ restricts to an automorphism of $K$ .
2	$σ$ sends $H$ to itself, and in fact restricts to an automorphism of $H$ .	definition of characteristic subgroup	$H$ is characteristic in $K$	Step (1)	direct
3	$c_{g}$ sends $H$ to itself and restricts to an automorphism of $H$ .			Steps (1), (2)	[SHOW MORE] Restricting a function from $G$ to $K$ and then from $K$ to $H$ is equivalent to directly restricting the function from $G$ to $H$ . By Step (1), $c_{g}$ on $G$ restricts to $σ$ on $K$ , which by Step (2) restricts to an automorphism of $H$ . Thus, $c_{g}$ also restricts to the same automorphism of $H$ .

Using function restriction expressions

In terms of the function restriction formalism:

The following is a function restriction expression for the subgroup property of normality:

Inner automorphism $\to$ Automorphism

In other words, every inner automorphism of the whole group restricts to an automorphism of the subgroup.

The following is a function restriction expression for the subgroup property of characteristicity:

Automorphism $\to$ Automorphism

In other words, every automorphism of the whole group restricts to an automorphism of the subgroup.

We now use the composition rule for function restriction to observe that the composition of characteristic and normal implies the property:

Inner automorphism $\to$ Automorphism

Which is again the subgroup property of normality.

References

Textbook references

Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347, Page 135, Section 4.4 (Automorphisms), Point (3) after definition of characteristic subgroup, ^{More info}. Also, Page 137, Exercise 8(a).
A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613, Page 28, Section 1.5 (Characteristic and Fully invariant subgroups), 1.5.6(iii), ^{More info}
Topics in Algebra by I. N. Herstein, Page 70, Problem 9, ^{More info}
Nilpotent groups and their automorphisms by Evgenii I. Khukhro, ISBN 3110136724, Page 4, Section 1.1, (passing mention)^{More info}

External links

Search links

Google Search

@@ Line 1: / Line 1: @@
-{{composition computations}}
+{{composition computation|
+left = characteristic subgroup|
+right = normal subgroup|
+final = normal subgroup}}
+[[difficulty level::1| ]]
 ==Statement==
@@ Line 11: / Line 14: @@
 ===Verbal statement===
-Every [[characteristic subgroup]] of a [[normal subgroup]] is [[normal subgroup|normal]].
+Every [[fact about::characteristic subgroup;2| ]][[characteristic subgroup]] of a [[fact about::normal subgroup;2| ]][[normal subgroup]] is [[normal subgroup|normal]].
-===Symbolic statement===
+===Statement with symbols===
-Let <math>H \le K \le G</math> such that <math>H</math> is characteristic in <math>K</math> and <math>K</math> is normal in <math>G</math>, then <math>H</math> is normal in <math>G</math>.
+Let <math>H \le K \le G</math> be groups such that <math>H</math> is characteristic in <math>K</math> and <math>K</math> is normal in <math>G</math>. Then, <math>H</math> is normal in <math>G</math>.
 ==Related facts==
+===Basic ideas implicit in the definitions===
+* [[Restriction of automorphism to subgroup invariant under it and its inverse is automorphism]]: If <math>K \le G</math> is a subgroup and <math>\sigma</math> is an automorphism of <math>G</math> such that both <math>\sigma</math> and <math>\sigma^{-1}</math> send <math>K</math> to within itself, then <math>\sigma</math> restricts to an automorphism of <math>K</math>. This is the key idea used in arguing that an inner automorphism of the biggest group must restrict to an ''automorphism'' of the intermediate subgroup, rather than merely to a homomorphism from the intermediate subgroup to itself. Note that this idea is implicit in the equivalence between different formulations of the notion of normal subgroup.
+===Related facts in group theory===
+* [[Characteristicity is transitive]]: A characteristic subgroup of a characteristic subgroup is characteristic.
 * [[Left transiter of normal is characteristic]]: Characteristicity is the weakest, or most ''general'' property, for which the above statement is true. This is made precise in the statement that characteristicity is the [[left transiter]] for normality.
 * [[Automorph-permutable of normal implies conjugate-permutable]]: This statement has many corollaries; for instance, [[2-subnormal implies conjugate-permutable]]
+===Analogues===
+* [[Derivation-invariant subring of ideal implies ideal]]: This is the analogous statement for [[Lie ring]]s. Here, [[derivation]]s play the role of automorphisms, [[Lie subring]]s play the role of [[subgroup]]s, [[ideal of a Lie ring|ideal]]s play the role of [[normal subgroup]]s, [[inner derivation]]s play the role of [[inner automorphism]]s, and [[derivation-invariant subring]]s play the role of [[characteristic subgroup]]s.
+==Applications==
+For a complete list of applications, refer:
+[[:Category:Applications of characteristic of normal implies normal]]
+==Definitions used==
+===Characteristic subgroup===
+{{further|[[Characteristic subgroup]]}}
+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>.
+* '''Definition using [[function restriction expression]]''': We can write characteristicity as the [[balanced subgroup property]] with respect to automorphisms:
+Characteristic = Automorphism <math>\to</math> Automorphism
+This is interpreted as: any automorphism from the whole group to itself, ''restricts'' to an automorphism from the subgroup to itself. Note that this is stronger than simply saying that it maps the subgroup to within itself -- we also demand that the restriction be an automorphism of the subgroup.
+===Normal subgroup===
+{{further|[[Normal subgroup]]}}
+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 a map from <math>H</math> to itself. In other words, for any <math>g \in G</math>:
+<math>c_g(H) \le H</math>
+or more explicitly:
+<math>gHg^{-1} \le H</math>
+Implicit in this definition is the fact that <math>c_g</math> ''is'' an automorphism. {{further|[[Group acts as automorphisms by conjugation]]}}
+Note that it turns out that the above also implies that <math>c_g(H) = H</math> (This is because we have <math>c_g(H) \le H</math> as well as <math>c_{g^{-1}}(H) \le H</math>). This equivalence of ideas is crucial to the proof.
+* ''Definition using [[function restriction expression]]'': We can write normality as the invariance property with respect to inner automorphisms:
+Normal = [[Inner automorphism]] <math>\to</math> Automorphism
+In other words, any inner automorphism on the whole group ''restricts'' to an automorphism from the subgroup to itself. Note that this is stronger than saying that the inner automorphism simply sends the subgroup to itself -- we also demand that the restriction itself be an automorphism of the subgroup.
+==Facts used==
+# [[uses::Restriction of automorphism to subgroup invariant under it and its inverse is automorphism]]
+# [[uses::Composition rule for function restriction]]: This is used for the proof using function restriction expressions.
 ==Proof==
+{{tabular proof format}}
 ===Hands-on proof===
-Given groups <math>H \le K \le G</math> such that <math>H</math> is characteristic in <math>K</math> and <math>K</math> is normal in <math>G</math>. We need to show that for any <math>g \in G</math>, the map <math>c_g : x \mapsto gxg^{-1}</math> takes <math>H</math> to within itself.
+'''Given''': Groups <math>H \le K \le G</math> such that <math>H</math> is characteristic in <math>K</math> and <math>K</math> is normal in <math>G</math>. An element <math>g \in G</math>.
-First, notice that since <math>K \triangleleft G</math>, <math>c_g(x) \in K</math> for every <math>x \in K</math>. Thus, <math>c_g</math> restricts to a function from <math>K</math> to <math>K</math>. Since this function arises by restricting an automorphism of <math>G</math>, it is an endomorphism of <math>K</math>.
+'''To Prove''': The map <math>g \in G</math>, the map <math>c_g : x \mapsto gxg^{-1}</math> maps <math>H</math> to <math>H</math> (and in fact, yields an automorphism of <math>H</math>).
-Further, since <math>c_{g^{-1}} \circ c_g</math> is the identity map, and <math>K</math> is invariant under both, the restriction of <math>c_g</math> to <math>H</math> is actually an invertible endomorphism, viz an automorphism. Call this automorphism <math>\sigma</math>.
+'''Proof''':
-Since <math>H</math> is characteristic in <math>K</math>, <math>\sigma</math> takes <math>H</math> to within itself. But since <math>\sigma</math> is the restriction of <math>c_g</math> to <math>K</math> in the first place, we conclude that <math>c_g</math> in fact takes <math>H</math> to itself.
+{| class="sortable" border="1"
+! Step no. !! Assertion/construction !! Facts used !! Given data used !! Previous steps used !! Explanation
+|-
+| 1 || <math>c_g(x) \in K</math> for every <math>x \in K</math> and <math>c_g</math> restricts to an automorphism of <math>K</math>. Call this automorphism <math>\sigma</math>. || definition of normal subgroup<br>Fact (1) || <math>K</math> is normal in <math>G</math><br><math>g</math> is in <math>G</math>. || || <toggledisplay>By the definition of normal, <math>c_g</math> sends <math>K</math> to <math>K</math>. So does <math>c_{g^{-1}}</math>. The two conjugation maps are inverses of each other, so by Fact (1), <math>c_g</math> restricts to an automorphism of <math>K</math>.</toggledisplay>
+|-
+| 2 || <math>\sigma</math> sends <math>H</math> to itself, and in fact restricts to an automorphism of <math>H</math>. || definition of characteristic subgroup || <math>H</math> is characteristic in <math>K</math> || Step (1) || direct
+|-
+| 3 || <math>c_g</math> sends <math>H</math> to itself and restricts to an automorphism of <math>H</math>. || || || Steps (1), (2) || <toggledisplay>Restricting a function from <math>G</math> to <math>K</math> and then from <math>K</math> to <math>H</math> is equivalent to directly restricting the function from <math>G</math> to <math>H</math>. By Step (1), <math>c_g</math> on <math>G</math> restricts to <math>\sigma</math> on <math>K</math>, which by Step (2) restricts to an automorphism of <math>H</math>. Thus, <math>c_g</math> also restricts to the same automorphism of <math>H</math>.</toggledisplay>
+|}
-===Using the function restriction formalism===
+===Using function restriction expressions===
 In terms of the function restriction formalism:
-* The following is a [[restriction formal expression]] for the subgroup property of normality:
+* The following is a [[function restriction expression]] for the subgroup property of normality:
 Inner automorphism <math>\to</math> Automorphism
-* The following is a [[restriction formal expression]] for the subgroup property of characteristicity:
+In other words, every inner automorphism of the whole group restricts to an automorphism of the subgroup.
+* The following is a [[function restriction expression]] for the subgroup property of characteristicity:
 Automorphism <math>\to</math> Automorphism
+In other words, every automorphism of the whole group restricts to an automorphism of the subgroup.
 We now use the [[composition rule for function restriction]] to observe that the composition of characteristic and normal implies the property:
@@ Line 54: / Line 130: @@
 ==References==
 ===Textbook references===
-* {{booklink|DummitFoote}}, page 135
+* {{booklink-stated|DummitFoote|135|Section 4.4 (''Automorphisms''), Point (3) after definition of characteristic subgroup}}. Also, [[Stated in::DummitFoote;137;Exercise 8(a)| ]] Page 137, Exercise 8(a).
+* {{booklink-proved|RobinsonGT|28| Section 1.5 (''Characteristic and Fully invariant subgroups''), 1.5.6(iii)}}
+* {{booklink-stated|Herstein|70|Problem 9}}
+* {{booklink-stated|KhukhroNGA|4|Section 1.1|passing mention}}
 ==External links==
 ===Search links===
 * [http://www.google.com/search?q=%22characteristic+subgroup+of+a+normal+subgroup+is+normal%22&ie=utf-8&oe=utf-8&aq=t&rls=org.mozilla:en-US:official&client=firefox-a| Google Search]