Left 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 left tight function restriction expression, i.e., if $a$ cannot be weakened further without changing $b$, and if $p$ is an identity-true subgroup property, i.e., every group has property $p$ as a subgroup of itself, then the right transiter for $p$ is the subgroup property $a\to a$.
• Otherwise, if $p$ is an identity-true subgroup property, let $c\to b$ be the left tightening of $a\to b$. Then, the right transiter of $p$ is the property $c\to c$.

Related facts

• Balanced implies transitive
• Right tightness theorem