Endo-invariance property

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Definition

Symbol-free definition

A subgroup property $p$ 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 $p$ in the group if and only if it is invariant under all the endomorphisms in that collection.

In terms of the function restriction formalism

A subgroup property $p$ is termed an endo-invariance property if it has a function restriction expression of the form:

$a \to$ Function

where $a$ is a property of endomorphisms (in other words, $a$ is a function property satisfied only by endomorphisms).

Equivalently, it can be expressed in the form:

$a \to$ Endomorphism

Equivalence of definitions

The equivalence of the various definitions follows from this observation: restriction of endomorphism to invariant subgroup is endomorphism.