Right tightness theorem

Statement

For function restriction expressions for subgroup properties

Suppose $a$ and $b$ are properties of functions from a group to itself, and $p$ is a subgroup property with the function restriction expression:

$a\to b$.

In other words, a subgroup $H$ satisfies property $p$ in a group $G$ if and only if every function from $G$ to itself satisfying property $a$ in $G$, restricts to a function from $H$ to itself satisfying property $b$ in $H$.

Then:

• If $a\to b$ is a right tight function restriction expression, i.e., if $b$ cannot be strengthened further without changing $a$, then the left transiter for $p$ is the subgroup property $b\to b$.
• Otherwise, compute the property $c$ such that $a\to c$ is a right tightening for $a\to b$. Then the left transiter of $p$ is the property $c\to c$.