Function property

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This article is about a general term. A list of important particular cases (instances) is available at Category:Function properties

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Definition

A function property is a map from the collection of all possible functions from a group to itself, to the two-element set (true, false). A function which gets mapped to true is said to have the function property, and a function which gets mapped to false is said to not have the function property.

The function property must satisfy isomorphism-invariance: if $f_{1} : G \to G$ and $f_{2} : H \to H$ are functions, and there is an isomorphism $σ : G \to H$ such that $σ \circ f_{1} = f_{2} \circ σ$ , then $f_{1}$ satisfies the function property iff $f_{2}$ satisfies the function property.