# Abelian-extensible endomorphism-invariant subgroup

From Groupprops

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This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

## Definition

A subgroup of an abelian group is termed an **abelian-extensible endomorphism-invariant subgroup** if it is invariant under all the abelian-extensible endomorphisms of .

## Formalisms

### Function restriction expression

This subgroup property is a function restriction-expressible subgroup property: it can be expressed by means of the function restriction formalism, viz there is a function restriction expression for it.

Find other function restriction-expressible subgroup properties | View the function restriction formalism chart for a graphic placement of this property

Function restriction expression | is a fully invariant subgroup of if ... | This means that full invariance is ... | Additional comments |
---|---|---|---|

abelian-extensible endomorphism function | every abelian-extensible endomorphism of sends every element of to within | the invariance property for abelian-extensible endomorphisms | |

abelian-extensible endomorphism endomorphism | every abelian-extensible endomorphism of restricts to an endomorphism of | the endo-invariance property for abelian-extensible endomorphisms; i.e., it is the invariance property for abelian-extensible endomorphism, which is a property stronger than the property of being an endomorphism |

## Relation with other properties

### Stronger properties

property | quick description | proof of implication | proof of strictness (reverse implication failure) | intermediate notions |
---|---|---|---|---|

Fully invariant subgroup of abelian group | ||||

Subgroup of finite abelian group |

### Weaker properties

property | quick description | proof of implication | proof of strictness (reverse implication failure) | intermediate notions |
---|---|---|---|---|

Subgroup of abelian group |