# Endo-invariance property

From Groupprops

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## Definition

### Symbol-free definition

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

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

Function

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.