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A subgroup property is termed an endo-invariance property if for any group, there is a collection of endomorphisms of the group such that a subgroup has property in the group if and only if it is invariant under all the endomorphisms in that collection.
In terms of the function restriction formalism
where is a property of endomorphisms (in other words, is a function property satisfied only by endomorphisms).
Equivalently, it can be expressed in the form:
Equivalence of definitions
The equivalence of the various definitions follows from this observation: restriction of endomorphism to invariant subgroup is endomorphism.