Extensibility-stable function property

This article defines a function metaproperty; a property that can be evaluated for properties of functions from a group to itself

Definition

Symbol-free definition

A property $p$ of functions from a group to itself is said to be extensibility-stable if, given any subgroup of a group, and a function on the subgroup satisfying property $p$ in the subgroup, there is a function on the group satisfying property $p$ that extends the given function.

Definition with symbols

A property $p$ of functions from a group to itself is said to be extensibility-stable if, given any groups $H\le G$, and a function $f:H\to H$ satisfying property $p$ in $H$, there is a function $f^{\prime }:G\to G$ such that $f^{\prime }$ satisfies $p$ and such that the restriction of $f^{\prime }$ to $H$ is $f$.

In terms of the extensibility operator

Extensibility-stable function properties are precisely those function properties that are fixedp oints underthe extensibility operator on the function property space.

Examples

Inner automorphism

Any inner automorphism of a subgroup lifts to an inner automorphism of the whole group. This is because we can take the same conjugating element from the subgroup and use it to define a conjugation on the whole group. Note that since for a given inner automorphism, the choice of conjugating element is not unique, the lift is in general not unique. For full proof, refer: Inner is extensibility-stable