Balanced implies transitive

Statement
Suppose $$p$$ is a balanced subgroup property with respect to the function restriction formalism. In other words, there exists a function property $$a$$ such that we can write:

$$p = a \to a$$.

Specifically, a subgroup $$H$$ of a group $$G$$ satisfies property $$p$$ in $$G$$ if and only if every function from $$G$$ to $$G$$ satisfying property $$a$$ restricts to a function from $$H$$ to $$H$$ satisfying property $$a$$.

Then, $$p$$ is a transitive subgroup property. In other words, if $$H \le K \le G$$ are subgroups such that $$H$$ has property $$p$$ in $$K$$ and $$K$$ has property $$p$$ in $$G$$, then $$H$$ has property $$p$$ in $$G$$.

Stronger facts

 * 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.

Converse
A converse of sorts is true:


 * Function restriction-expressible and t.i. implies balanced
 * Left tightness theorem
 * Right tightness theorem

Particular cases

 * Characteristicity is transitive
 * Central factor is transitive
 * Full invariance is transitive
 * Injective endomorphism-invariance is transitive

Function restriction expression
Suppose $$a,b$$ are properties of functions from a group to itself. Then, the property:

$$p := a \to b$$

is defined as follows. A subgroup $$H$$ of a group $$G$$ has property $$p$$ in $$G$$ if whenever $$f: G \to G$$ is a function satisfying property $$a$$, $$f$$ restricts to a function from $$H$$ to $$H$$ and the restriction of $$f$$ to $$H$$ satisfies property $$b$$.

Balanced subgroup property
A subgroup property $$p$$ is termed balanced with respect to the function restriction formalism if there exists a function property $$a$$ such that:

$$p = a \to a$$.

Composition operator
Suppose $$p, q$$ are subgroup properties. The composition of these properties, denoted $$p * q$$, is defined as follows: A subgroup $$H$$ of a group $$G$$ is said to have property $$p*q$$ in $$G$$ if and only if there exists a subgroup $$K$$ such that $$H \le K \le G$$, and such that $$H$$ has property $$p$$ in $$K$$ and $$K$$ has property $$q$$ in $$G$$.

Examples
Here are some examples of balanced subgroup properties that are also therefore transitive:


 * Fully characteristic subgroup: The function restriction expression for this is:

Endomorphism $$\to$$ Endomorphism.

This is balanced. Thus, the property of being fully characteristic is transitive.


 * Characteristic subgroup: The function restriction expression for this is:

Automorphism $$\to$$ Automorphism.

Thus, the property of being a characteristic subgroup is transitive.


 * Central factor: The function restriction expression for this is:

Inner automorphism $$\to$$ [[Inner automorphism..

Thus, the property of being a central factor is transitive.

For more examples of balanced subgroup properties, view:

Category:Balanced subgroup properties

Each of these is transitive.

Facts used

 * 1) uses::Composition rule for function restriction:If $$a, b, c, d$$ are function properties such that $$b \le c$$, then:

$$(c \to d) * (a \to b) \le a \to d$$

Here, $$*$$ denotes the composition operator for subgroup properties.

Proof
Given: A balanced subgroup property $$p = a \to a$$.

To prove: $$p * p \le p$$.

Proof: Apply fact (1) with all four variables set to equal $$a$$. We get:

$$(a \to a) * (a \to a) \le a \to a$$.

This yields the required result:

$$p * p \le p$$.