Balanced subgroup property (function restriction formalism): Difference between revisions

Latest revision as of 02:18, 12 January 2026

This article defines a subgroup metaproperty: a property that can be evaluated to true/false for any subgroup property
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This article is about a general term. A list of important particular cases (instances) is available at Category:Balanced subgroup properties

Definition

Symbol-free definition

A subgroup property is said to be a balanced subgroup property if it can be expressed via a function restriction expression with both the left side and the right side being equal.

Definition with symbols

A subgroup property is said to be a balanced subgroup property if it can be expressed as $a \to a$ where $a$ is a function property. In other words, a subgroup $H$ satisfies the property in a group $G$ if and only if every function from $G$ to itself satisfying property $a$ restricts to a function from $H$ to itself satisfying property $a$ .

Examples

Subgroup property	Function property for which it is the balanced subgroup property	Function restriction expression	Further comments
characteristic subgroup	automorphism	automorphism $\to$ automorphism	The right side can be weakened all the way to function, i.e., characteristicity can be written as automorphism $\to$ function. In other words, it is the invariance property with respect to automorphisms.
fully invariant subgroup	endomorphism	endomorphism $\to$ endomorphism	The right side can be weakened all the way to function, i.e., full invariance can be written as endomorphism $\to$ function. In other words, it is the invariance property with respect to endomorphisms.
central factor	inner automorphism	inner automorphism $\to$ inner automorphism
powering-invariant subgroup	rational power map	rational power map $\to$ rational power map	The right side can be weakened all the way to function, i.e., powering-invariance can be written as rational power map $\to$ function. In other words, it is the invariance property with respect to rational power maps.
local powering-invariant subgroup	local powering	local powering $\to$ local powering	The right side can be weakened all the way to function, i.e., local powering-invariance can be written as local powering $\to$ function. In other words, it is the invariance property with respect to local powerings.

Relation with other metaproperties

T.i. subgroup properties

Clearly, any balanced subgroup property with respect to the function restriction formalism is both transitive and identity-true. Hence, it is a t.i. subgroup property.

Interestingly, a partial converse holds by the balance theorem: every t.i. subgroup property that can be expressed using the function restriction formalism, is actually a balanced subgroup property. In fact, more strongly, a balanced expression for the property can be obtained by using either the right tightening operator or the left tightening operator to any starting expression.

Intersection-closedness

In general, a balanced subgroup property need not be intersection-closed.

Join-closedness

In general, a balanced subgroup property need not be join-closed.

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 ===Definition with symbols===
-A [[subgroup property]] is said to be a '''balanced subgroup property''' if it can be expressed as <math>a \to a</math> where <math>a</math> is a function property. In other words, a subgroup <math>H</math> satisfies the property in a group <math>G</math> if and only if every function on <math>G</math> satisfying property <math>a</math> in <math>G</math> restricts to a function satisfying property <math>a</math> on <math>H</math>.
+A [[subgroup property]] is said to be a '''balanced subgroup property''' if it can be expressed as <math>a \to a</math> where <math>a</math> is a function property. In other words, a subgroup <math>H</math> satisfies the property in a group <math>G</math> if and only if every function from <math>G</math> to itself satisfying property <math>a</math> restricts to a function from <math>H</math> to itself satisfying property <math>a</math>.
 ==Examples==
-===Characteristic subgroup===
+{| class="sortable" border="1"
+! Subgroup property !! Function property for which it is the balanced subgroup property !! Function restriction expression !! Further comments
-The property of a subgroup being [[characteristic subgroup|characteristic]] is expressible as a balanaced subgroup property in the function restriction formalism as follows:
+|-
+| [[characteristic subgroup]] || [[automorphism]] || [[automorphism]] <math>\to</math> [[automorphism]] || The right side can be weakened all the way to function, i.e., characteristicity can be written as [[automorphism]] <math>\to</math> function. In other words, it is the [[invariance property]] with respect to automorphisms.
-Automorphism <math>\to</math> Automorphism
+|-
+| [[fully invariant subgroup]] || [[endomorphism]] || [[endomorphism]] <math>\to</math> [[endomorphism]] || The right side can be weakened all the way to function, i.e., full invariance can be written as [[endomorphism]] <math>\to</math> function. In other words, it is the [[invariance property]] with respect to endomorphisms.
-===Other examples===
+|-
+| [[central factor]] || [[inner automorphism]] || [[inner automorphism]] <math>\to</math> [[inner automorphism]] ||
-* [[Fully characteristic subgroup]] = [[Endomorphism]] <math>\to</math> endomorphism
+|-
-* [[I-characteristic subgroup]] = Injective endomorphism <math>\to</math> injective endomorphism
+| [[powering-invariant subgroup]] || rational power map || rational power map <math>\to</math> rational power map || The right side can be weakened all the way to function, i.e., powering-invariance can be written as rational power map <math>\to</math> function. In other words, it is the [[invariance property]] with respect to rational power maps.
-* [[Retraction-invariant subgroup]] = Retraction <math>\to</math> Retraction
+|-
-* [[Transitively normal subgroup]] = [[Normal automorphism]] <math>\to</math> Normal automorphism
+| [[local powering-invariant subgroup]] || local powering || local powering <math>\to</math> local powering || The right side can be weakened all the way to function, i.e., local powering-invariance can be written as local powering <math>\to</math> function. In other words, it is the [[invariance property]] with respect to local powerings.
-* [[Conjugacy-closed normal subgroup]] = [[Class automorphism]] <math>\to</math> Class automorphism
+|}
-* [[Central factor]] = [[Inner automorphism]] <math>\to</math> inner automorphism
 ==Relation with other metaproperties==