Balanced subgroup property (function restriction formalism)

From Groupprops
(Redirected from Balanced subgroup property)
Jump to: navigation, search
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 metaproperty
VIEW 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 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

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.