Function property

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 $$\sigma:G \to H$$ such that $$\sigma \circ f_1 = f_2 \circ \sigma$$, then $$f_1$$ satisfies the function property iff $$f_2$$ satisfies the function property.