Balanced implies transitive
This article gives the statement and possibly, proof, of an implication relation between two subgroup metaproperties. That is, it states that every subgroup satisfying the first subgroup metaproperty (i.e., Balanced subgroup property (function restriction formalism) (?)) must also satisfy the second subgroup metaproperty (i.e., Transitive subgroup property (?))
View all subgroup metaproperty implications | View all subgroup metaproperty non-implications
Suppose is a balanced subgroup property with respect to the function restriction formalism. In other words, there exists a function property such that we can write:
Specifically, a subgroup of a group satisfies property in if and only if every function from to satisfying property restricts to a function from to satisfying property .
Then, is a transitive subgroup property. In other words, if are subgroups such that has property in and has property in , then has property in .
- Balanced implies t.i.: A balanced subgroup property is a t.i. subgroup property: it is transitive as well as identity-true: every group has the property as a subgroup of itself.
A converse of sorts is true:
- Function restriction-expressible and t.i. implies balanced
- Left tightness theorem
- Right tightness theorem
- Characteristicity is transitive
- Central factor is transitive
- Full invariance is transitive
- Injective endomorphism-invariance is transitive
Function restriction expression
Further information: function restriction expression
Suppose are properties of functions from a group to itself. Then, the property:
is defined as follows. A subgroup of a group has property in if whenever is a function satisfying property , restricts to a function from to and the restriction of to satisfies property .
Balanced subgroup property
Further information: Balanced subgroup property (function restriction formalism)
A subgroup property is termed balanced with respect to the function restriction formalism if there exists a function property such that:
Further information: composition operator
Suppose are subgroup properties. The composition of these properties, denoted , is defined as follows: A subgroup of a group is said to have property in if and only if there exists a subgroup such that , and such that has property in and has property in .
Here are some examples of balanced subgroup properties that are also therefore transitive:
- Fully invariant subgroup: The function restriction expression for this is:
This is balanced. Thus, the property of being fully characteristic is transitive. For full proof, refer: Full invariance is transitive
- Characteristic subgroup: The function restriction expression for this is:
Thus, the property of being a characteristic subgroup is transitive. For full proof, refer: Characteristicity is transitive
- Central factor: The function restriction expression for this is:
Thus, the property of being a central factor is transitive. For full proof, refer: Central factor is transitive
- Composition rule for function restriction:If are function properties such that , then:
Here, denotes the composition operator for subgroup properties.
Given: A balanced subgroup property .
To prove: .
Proof: Apply fact (1) with all four variables set to equal . We get:
This yields the required result: