Proving transitivity

This is a survey article related to:subgroup metaproperty satisfaction
View other survey articles about subgroup metaproperty satisfaction

A subgroup property $p$ is termed a transitive subgroup property if whenever $H\leq K\leq G$ are groups such that $H$ satisfies property $p$ in $K$ and $K$ satisfies property $p$ in $G$, then $H$ satisfies property $p$ in $G$.

A subgroup property $p$ is termed a t.i. subgroup property if it is transitive as well as identity-true: every group satisfies the property as a subgroup of itself.

This article discusses techniques to prove that a given subgroup property is transitive. To learn about techniques to prove that a given subgroup property is not transitive, refer the survey article disproving transitivity.

Quick discussion on transitivity and subordination

Given a subgroup property $p$, we define the subordination of $p$ as the following property: $H$ has the property in $G$ if there exists an ascending chain of subgroups:

$H=H_{0}\leq H_{1}\leq H_{2}\leq \dots \leq H_{n}=G$,

such that each $H_{i-1}$ satisfies property $p$ in $H_{i}$. By definition, any group satisfies the subordination of $p$ in itself, since we can take $n=0$ and take a chain of length $0$.

The subordination of any property is a t.i. subgroup property (it is both transitive and identity-true) and a t.i. subgroup property equals its own subordination.

Here are some quick points on the subordination operator:

1. The subordination operator is an ascendant operator: If $p$ is a subgroup property, $p$ is stronger than its subordination.
2. The subordination operator is a monotone operator: If $p,q$ are subgroup properties such that $p$ is stronger than $q$, then the subordination of $p$ is stronger than the subordination of $q$.
3. If $p$ and $q$ are subgroup properties such that $q$ is t.i., then the subordination of $p$ is stronger than $q$. In fact, the subordination of $p$ is the strongest t.i. subgroup property among those weaker than $q$.

The basic proof idea: express the subgroup property in a form that makes it obvious

The idea is to express the subgroup property using a formalism that makes it obvious that it is transitive.

The most typical idea is that of a balanced subgroup property. We discuss the idea for function restriction expressions first, and then discuss some other, more general, variants.

Balanced subgroup properties in the function restriction formalism

Further information: Balanced subgroup property (function restriction formalism), balanced implies transitive

Suppose $a$ and $b$ are properties of functions from a group to itself. The property $p$ with function restriction expression $a\to b$ is defined as follows: $H$ satisfies $p$ in $G$ if every function from $G$ to itself satisfying $a$ restricts to a function from $H$ to itself satisfying $b$. (A bunch of examples is available at the function restriction formalism chart).

A balanced subgroup property with respect to the function restriction formalism is a subgroup property having an expression where both the left and right sides are equal. Such an expression is termed a balanced expression. For instance:

• The property of being a characteristic subgroup is a balanced subgroup property, because it can be expressed as:

Automorphism $\to$ Automorphism

• The property of being a central factor is a balanced subgroup property, because it can be expressed as:

Inner automorphism $\to$ Inner automorphism

• The property of being a fully characteristic subgroup is a balanced subgroup property, because it can be expressed as:

Endomorphism $\to$ Endomorphism

The easy but important fact is that any balanced subgroup property is transitive. In fact, for properties that have function restriction expressions, being t.i. (transitive and identity-true) is equivalent to being balanced, something that follows from either the left tightness theorem or the right tightness theorem. Note that a transitive subgroup property may have another function restriction expression that is not balanced; however, either left tightening or right tightening yields a balanced expression.

The idea of balance in other formalisms

For any formalism that involves restricting/extending functions, relations, or other constructs, the properties having balanced expressions are transitive. Here are some examples:

1. Function extension expressions: A function extension expression $a\leftarrow b$ defines a property as follows: $H$ has the property in $G$ if every function on $H$ satisfying property $b$ on $H$ can be extended to a function on $G$ satisfying property $a$ on $G$. A balanced expression is where $a=b$. Examples include:
   • AEP-subgroup, where both sides are automorphism.
   • EEP-subgroup, where both sides are endomorphism.

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