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 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 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 .
The property of a subgroup being characteristic is expressible as a balanaced subgroup property in the function restriction formalism as follows:
- 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
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.
In general, a balanced subgroup property need not be intersection-closed.
In general, a balanced subgroup property need not be join-closed.