# Injective endomorphism-invariant subgroup

From Groupprops

BEWARE!This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

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]

If the ambient group is a finite group, this property is equivalent to the property:characteristic subgroup

View other properties finitarily equivalent to characteristic subgroup | View other variations of characteristic subgroup | Read a survey article on varying characteristic subgroup

## Definition

QUICK PHRASES: invariant under all injective endomorphisms, injective endomorphism-invariant

### Equivalent definitions in tabular format

Below are many **equivalent** definitions of injective endomorphism-invariant subgroup.

No. | Shorthand | A subgroup of a group is characteristic in it if... | A subgroup of a group is called a characteristic subgroup of if ... |
---|---|---|---|

1 | injective endomorphism-invariant | every injective endomorphism of the whole group takes the subgroup to within itself | for every injective endomorphism of , . More explicitly, for any and that is injective, |

2 | injective endomorphisms restrict to endomorphisms | every injective endomorphism of the group restricts to an endomorphism of the subgroup. | for every injective endomorphism of , and restricts to an endomorphism of |

3 | injective endomorphisms restrict to injective endomorphisms | every injective endomorphism of the group restricts to an injective endomorphism of the subgroup | for every injective endomorphism of , and restricts to an injective endomorphism of |

This definition is presented using a tabular format. |View all pages with definitions in tabular format

## 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 an injective endomorphism-invariant subgroup of if ... | This means that injective endomorphism-invariance is ... | Additional comments |
---|---|---|---|

injective endomorphism function | every injective endomorphism of sends every element of to within | the invariance property for injective endomorphisms | |

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

injective endomorphism injective endomorphism | every injective endomorphism of restricts to a injective endomorphism of | the balanced subgroup property for injective endomorphisms | Hence, it is a t.i. subgroup property, both transitive and identity-true |

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

fully invariant subgroup | |FULL LIST, MORE INFO | |||

isomorph-free subgroup | Intermediately injective endomorphism-invariant subgroup, Isomorph-containing subgroup|FULL LIST, MORE INFO | |||

isomorph-containing subgroup | Intermediately injective endomorphism-invariant subgroup|FULL LIST, MORE INFO | |||

intermediately injective endomorphism-invariant subgroup | |FULL LIST, MORE INFO |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

characteristic subgroup | invariant under all automorphisms | injective endomorphism-invariant implies characteristic | characteristic not implies injective endomorphism-invariant | |FULL LIST, MORE INFO |

normal subgroup | invariant under all inner automorphisms | (via characteristic) | (via characteristic) | Characteristic subgroup|FULL LIST, MORE INFO |

### Related properties

- Strictly characteristic subgroup: This is the invariance property with respect to surjective, rather than injective, endomorphisms.

## Metaproperties

BEWARE!This section of the article uses terminology local to the wiki, possibly without giving a full explanation of the terminology used (though efforts have been made to clarify terminology as much as possible within the particular context)

Here is a summary:

Metaproperty name | Satisfied? | Proof | Difficulty level (0-5) | Statement with symbols |
---|---|---|---|---|

transitive subgroup property | Yes | follows from balanced implies transitive | If are groups such that is injective endomorphism-invariant in and is injective endomorphism-invariant in , then is injective endomorphism-invariant in . | |

trim subgroup property | Yes | Obvious reasons | 0 | In any group , the trivial subgroup and the whole group are injective endomorphism-invariant in |

strongly intersection-closed subgroup property | Yes | follows from invariance implies strongly intersection-closed | If , are all injective endomorphism-invariant in , so is the intersection of subgroups . | |

strongly join-closed subgroup property | Yes | follows from endo-invariance implies strongly join-closed | If , are all injective endomorphism-invariant in , so is the join of subgroups |