Extensibility-stable function property

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

Contents

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':G \to G such that f' satisfies p and such that the restriction of f' 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

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