Left transiter of normal is characteristic: Difference between revisions

Latest revision as of 09:24, 8 January 2026

Template:Left transiter computation

Statement

Statement with symbols

Let $H \leq K$ be a subgroup. The following are equivalent:

$H$ is a characteristic subgroup of $K$ .
If $G$ is a group containing $K$ such that $K$ is a normal subgroup of $G$ , then $H$ is also a normal subgroup of $G$ .

Property-theoretic statement

The left transiter of the subgroup property of normality is the subgroup property of characteristicity. In other words:

Characteristic $*$ Normal $\leq$ Normal

Every characteristic subgroup of a normal subgroup is normal (the $*$ here is for the composition operator).

If $p$ is such that:

$p *$ Normal $\leq$ Normal

Then $p \leq$ Characteristic

Definitions used

Characteristic subgroup

Hands-on definition: A subgroup $H$ of a group $G$ is termed characteristic in $G$ if for any automorphism $σ$ of $G$ , $σ (H) = H$ .
Definition using function restriction expression: A subgroup $H$ of a group $G$ is termed characteristic in $G$ if it has the following function restriction expression:

Automorphism $\to$ Automorphism

In other words, every automorphism of $G$ restricts to an automorphism of $H$ .

Normal subgroup

Hands-on definition: A subgroup $H$ of a group $G$ is termed normal in $G$ if for every $g \in G$ , $g H g^{- 1} = H$ .
Definition using function restriction expression: A subgroup $H$ of a group $G$ is termed normal in $G$ if it has the following function restriction expression:

Inner automorphism $\to$ Automorphism

In other words, every inner automorphism of $G$ restricts to an automorphism of $H$ .

Related facts

Related left residual computations

All these use the fact that inner automorphism to automorphism is right tight for normality.

Some examples with a somewhat different flavor:

Left residual of normal by normal Hall is coprime automorphism-invariant normal

Left transiters of other closely related properties

Upper hooks and related facts

Normal upper-hook fully normalized implies characteristic: If $H \leq K \leq G$ are such that $H$ is normal in $G$ and $K$ is fully normalized in $G$ , then $H$ is characteristic in $K$ .
Characteristically simple and normal fully normalized implies minimal normal

Facts used

Characteristic of normal implies normal (used for the more direct part of the proof).
Inner automorphism to automorphism is right tight for normality
Left transiter of property with right tight function restriction expression is balanced property for right side

Proof

Hands-on proof

Forward direction ((1) implies (2))

This follows directly from Fact (1).

Reverse direction ((2) implies (1))

Given: A subgroup $H$ of a group $K$ such that for every group $G$ containing $K$ as a normal subgroup, $H$ is a normal subgroup of $G$ . An automorphism $σ$ of $K$ (i.e., $σ \in A u t (K)$ ).

To prove: $σ (H) = H$ .

Proof:

Step no.	Assertion/construction	Given data used	Previous steps used	Explanation
1	Let $G = H o l (K)$ denote the holomorph of $K$ . $H o l (K)$ is the semidirect product of $K$ with $A u t (K)$ .
2	$K$ is normal in $G$ .		Step (1)	This follows from the definition of semidirect product.
3	There exists an element $g \in G$ such that $σ$ is the restriction to $K$ of conjugation by $g$ (that we denote $c_{g}$ ).	$σ \in A u t (K)$	Step (1)	This follows from the definition of semidirect product.
4	$H$ is normal in $G$ .	$H$ is normal in any group containing $K$ as a normal subgroup.	Step (2)	Step/given data-combination direct
5	$c_{g} (H) = H$ .		Steps (3), (4)	Step-combination direct
6	$σ (H) = H$ .		Steps (3), (5)	Step-combination direct

Property-theoretic proof

By fact (2), the function restriction expression:

Inner automorphism $\to$ Automorphism

is a right tight function restriction expression for normality. In other words, for every automorphism of a group, there is a bigger group in which it is normal, such that the automorphism extends to an inner automorphism in that bigger group. Combining this with fact (3), we see that the left transiter of normality is the property:

Automorphism $\to$ Automorphism

which is indeed the property of being a characteristic subgroup.

@@ Line 1: / Line 1: @@
 {{left transiter computation}}
+[[difficulty level::3| ]]
 ==Statement==
-===Symbolic statement===
+===Statement with symbols===
 Let <math>H \le K</math> be a [[subgroup]]. The following are equivalent:
-# <math>H</math> is a [[fact about::characteristic subgroup]] of <math>K</math>
+# <math>H</math> is a [[fact about::characteristic subgroup;2| ]][[characteristic subgroup]] of <math>K</math>.
-# If <math>G</math> is a group containing <math>K</math> such that <math>K</math> is a [[fact about::normal subgroup]] of <math>G</math>, then <math>H</math> is also a normal subgroup of <math>G</math>.
+# If <math>G</math> is a group containing <math>K</math> such that <math>K</math> is a [[fact about::normal subgroup;2| ]][[normal subgroup]] of <math>G</math>, then <math>H</math> is also a normal subgroup of <math>G</math>.
 ===Property-theoretic statement===
@@ Line 23: / Line 23: @@
 Then <math>p \le</math> Characteristic
+==Definitions used==
+===Characteristic subgroup===
+* '''Hands-on definition''': A subgroup <math>H</math> of a group <math>G</math> is termed characteristic in <math>G</math> if for any automorphism <math>\sigma</math> of <math>G</math>, <math>\sigma(H) = H</math>.
+* '''Definition using [[function restriction expression]]''': A subgroup <math>H</math> of a group <math>G</math> is termed characteristic in <math>G</math> if it has the following function restriction expression:
+Automorphism <math>\to</math> Automorphism
+In other words, every automorphism of <math>G</math> restricts to an automorphism of <math>H</math>.
+===Normal subgroup===
+* '''Hands-on definition''': A subgroup <math>H</math> of a group <math>G</math> is termed normal in <math>G</math> if for every <math>g \in G</math>, <math>gHg^{-1} = H</math>.
+* '''Definition using [[function restriction expression]]''': A subgroup <math>H</math> of a group <math>G</math> is termed normal in <math>G</math> if it has the following function restriction expression:
+[[Inner automorphism]] <math>\to</math> Automorphism
+In other words, every inner automorphism of <math>G</math> restricts to an automorphism of <math>H</math>.
+==Related facts==
+===Related left residual computations===
+All these use the fact that [[inner automorphism to automorphism is right tight for normality]].
+* [[Left residual of conjugate-permutable by normal is automorph-permutable]]
+* [[Left residual of pronormal by normal is procharacteristic]]
+* [[Left residual of weakly pronormal by normal is weakly procharacteristic]]
+* [[Left residual of paranormal by normal is paracharacteristic]]
+* [[Left residual of polynormal by normal is polycharacteristic]]
+* [[Left residual of weakly normal by normal is weakly characteristic]]
+Some examples with a somewhat different flavor:
+* [[Left residual of normal by normal Hall is coprime automorphism-invariant normal]]
+===Left transiters of other closely related properties===
+* [[Characteristic implies left-transitively 2-subnormal]], [[Characteristic implies left-transitively fixed-depth subnormal]]
+* [[Subgroup-coprime automorphism-invariant implies left-transitively 2-subnormal]]
+* [[Complemented normal is not transitive]]
+* [[Left-transitively complemented normal implies characteristic]], [[Characteristic not implies left-transitively complemented normal]]
+* [[Left-transitively permutable implies characteristic]]
+===Upper hooks and related facts===
+* [[Normal upper-hook fully normalized implies characteristic]]: If <math>H \le K \le G</math> are such that <math>H</math> is normal in <math>G</math> and <math>K</math> is [[fully normalized subgroup|fully normalized]] in <math>G</math>, then <math>H</math> is characteristic in <math>K</math>.
+* [[Characteristically simple and normal fully normalized implies minimal normal]]
+==Facts used==
+# [[uses::Characteristic of normal implies normal]] (used for the more direct part of the proof).
+# [[uses::Inner automorphism to automorphism is right tight for normality]]
+# [[uses::Left transiter of property with right tight function restriction expression is balanced property for right side]]
 ==Proof==
-===1 implies 2===
+===Hands-on proof===
+====Forward direction ((1) implies (2))====
-{{proofat|[[Characteristic of normal implies normal]]}}
+This follows directly from Fact (1).
-To prove that <math>H</math> being characteristic in <math>K</math> implies the second condition, we need to show that if we start off with any inner automorphism of <math>G</math>, it leaves <math>H</math> invariant. We do this by restricting, first to <math>K</math>, and then from <math>K</math> to <math>H</math>.
+====Reverse direction ((2) implies (1))====
-===2 implies 1===
-We first sketch a hands-on proof, and then discuss the more general idea.
+'''Given''': A subgroup <math>H</math> of a group <math>K</math> such that for every group <math>G</math> containing <math>K</math> as a normal subgroup, <math>H</math> is a normal subgroup of <math>G</math>. An automorphism <math>\sigma</math> of <math>K</math> (i.e., <math>\sigma \in \operatorname{Aut}(K)</math>).
-For the group <math>K</math>, let <math>Hol(K)</math> denote the [[holomorph of a group|holomorph]] of <math>K</math>. <math>Hol(K)</math> is the semidirect product of <math>K</math> with <math>Aut(K)</math>. Observe that every automorphism of <math>K</math> lifts to an [[inner automorphism]] of <math>Hol(K)</math>, and further, that <math>K</math> is a normal subgroup of <math>Hol(K)</math>.
+'''To prove''': <math>\sigma(H) = H</math>.
-Now let <math>H</math> be a subgroup of <math>K</math> with the property that whenever <math>K</math> is normal in a group <math>G</math>, so is <math>H</math>. We will s how that <math>H</math> is characteristic in <math>K</math>.
+'''Proof''':
-Take <math>G = Hol(K)</math>. Clearly <math>K</math> is normal in <math>G</math>. Now, let <math>\sigma \in Aut(K)</math>. Clearly, there is an inner automorphism of <math>G</math> whose restriction to <math>K</math> is <math>\sigma</math>. But since every inner automorphism of <math>G</math> must leave <math>H</math> invariant, and hence <math>\sigma</math> must leave <math>H</math> invariant. This shows that <math>H</math> is invariant under all automorphisms of <math>K</math>, and hence, is a characteristic subgroup.
+{| class="sortable" border="1"
+! Step no. !! Assertion/construction !! Facts used !! Given data used !! Previous steps used !! Explanation
+|-
+| 1 || Let <math>G = \operatorname{Hol}(K)</math> denote the [[holomorph of a group|holomorph]] of <math>K</math>. <math>\operatorname{Hol}(K)</math> is the semidirect product of <math>K</math> with <math>\operatorname{Aut}(K)</math>.  || || || ||
+|-
+| 2 || <math>K</math> is normal in <math>G</math>. || || || Step (1) || This follows from the definition of semidirect product.
+|-
+| 3 || There exists an element <math>g \in G</math> such that <math>\sigma</math> is the restriction to <math>K</math> of conjugation by <math>g</math> (that we denote <math>c_g</math>). || || <math>\sigma \in \operatorname{Aut}(K)</math> || Step (1) || This follows from the definition of semidirect product.
+|-
+| 4 || <math>H</math> is normal in <math>G</math>. || || <math>H</math> is normal in any group containing <math>K</math> as a normal subgroup. || Step (2) || Step/given data-combination direct
+|-
+| 5 || <math>c_g(H) = H</math>. || || || Steps (3), (4) || Step-combination direct
+|-
+| 6 || <math>\sigma(H) = H</math>. || || || Steps (3), (5) || Step-combination direct
+|}
 ===Property-theoretic proof===
-To understand the proof property-theoretically, let us look at the [[function restriction expression]] for normality:
+By fact (2), the function restriction expression:
 Inner automorphism <math>\to</math> Automorphism
-It turns out that this function restriction expression for normality is [[right tight function restriction expression|right tight]]. In other words, we cannot replace ''Automorphism'' on the right by any stronger property. Equivalently, ''every'' automorphism ''can'' be realized as the restriction of an inner automorphism of a bigger group, for some embedding as a normal subgroup.
+is a right tight function restriction expression for normality. In other words, for ''every'' automorphism of a group, there is a bigger group in which it is normal, such that the automorphism extends to an inner automorphism in that bigger group. Combining this with fact (3), we see that the left transiter of normality is the property:
-(In fact, that is exactly what the [[holomorph]] construction above shows).
-Now, the [[transiter master theorem]] states that if <math>a \to b</math> is a [[right tight function restriction expression]] for a subgroup property then the left transiter of that subgroup property is <math>b \to b</math>. Thus, the left transiter of normality is:
 Automorphism <math>\to</math> Automorphism
-which is precisely the subgroup property of being characteristic.
+which is indeed the property of being a [[characteristic subgroup]].