# Balanced subgroup property (function restriction formalism)

This article defines a subgroup metaproperty: a property that can be evaluated to true/false for any subgroup property

View a complete list of subgroup metaproperties

View subgroup properties satisfying this metaproperty| View subgroup properties dissatisfying this metapropertyVIEW RELATED: subgroup metaproperty satisfactions| subgroup metaproperty dissatisfactions

BEWARE!This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

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 where is a function property. In other words, a subgroup satisfies the property in a group if and only if every function on satisfying property in restricts to a function satisfying property on .

## 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 Automorphism

### Other examples

- Fully characteristic subgroup = Endomorphism endomorphism
- Injective endomorphism-invariant subgroup = Injective endomorphism injective endomorphism
- Retraction-invariant subgroup = Retraction Retraction
- Transitively normal subgroup = Normal automorphism Normal automorphism
- Conjugacy-closed normal subgroup = Class automorphism Class automorphism
- Central factor = Inner automorphism 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.