Function property

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