Balanced subgroup property (function restriction formalism)

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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 on $G$ satisfying property $a$ in $G$ restricts to a function satisfying property $a$ on $H$.

Examples

Characteristic subgroup

The property of a subgroup being characteristic is expressible as a balanaced subgroup property in the function restriction formalism as follows:

Automorphism $\to$ Automorphism

Other examples

• Fully characteristic subgroup = Endomorphism $\to$ endomorphism
• Injective endomorphism-invariant subgroup = Injective endomorphism $\to$ injective endomorphism
• Retraction-invariant subgroup = Retraction $\to$ Retraction
• Transitively normal subgroup = Normal automorphism $\to$ Normal automorphism
• Conjugacy-closed normal subgroup = Class automorphism $\to$ Class automorphism
• Central factor = Inner automorphism $\to$ inner automorphism

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.