# Difference between revisions of "Injective endomorphism-invariant subgroup"

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{{quick phrase|[[quick phrase::invariant under all injective endomorphisms]], [[quick phrase::injective endomorphism-invariant]]}} | {{quick phrase|[[quick phrase::invariant under all injective endomorphisms]], [[quick phrase::injective endomorphism-invariant]]}} | ||

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− | + | ===Equivalent definitions in tabular format=== | |

− | + | Below are many '''equivalent''' definitions of injective endomorphism-invariant subgroup. | |

− | A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is | + | {| class="sortable" border="1" |

+ | ! No. !! Shorthand !! A [[subgroup]] of a [[group]] is characteristic in it if... !! A subgroup <math>H</math> of a group <math>G</math> is called a characteristic subgroup of <math>G</math> if ... | ||

+ | |- | ||

+ | | 1 || injective endomorphism-invariant || every [[injective endomorphism]] of the whole group takes the subgroup to within itself || for every [[injective endomorphism]] <math>\varphi</math> of <math>G</math>, <math>\varphi(H) \subseteq H</math>. More explicitly, for any <math>h \in H</math> and <math>\varphi \in \operatorname{End}(G)</math> that is injective, <math>\varphi(h) \in H</math> | ||

+ | |- | ||

+ | | 2 || injective endomorphisms restrict to endomorphisms || every injective endomorphism of the group restricts to an [[endomorphism]] of the subgroup. || for every injective endomorphism <math>\varphi</math> of <math>G</math>, <math>\varphi(H) \subseteq H</math> and <math>\varphi</math> restricts to an [[endomorphism]] of <math>H</math> | ||

+ | |- | ||

+ | | 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 <math>\varphi</math> of <math>G</math>, <math>\varphi(H) = H</math> and <math>\varphi</math> restricts to an [[injective endomorphism]] of <math>H</math> | ||

+ | |} | ||

+ | {{tabular definition format}} | ||

==Formalisms== | ==Formalisms== |

## Revision as of 00:48, 18 March 2019

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

- Fully characteristic subgroup
- Isomorph-free subgroup
- Isomorph-containing subgroup
- Intermediately injective endomorphism-invariant subgroup

### Weaker properties

- Characteristic subgroup:
*For proof of the implication, refer injective endomorphism-invariant implies characteristic and for proof of its strictness (i.e. the reverse implication being false) refer Characteristic not implies injective endomorphism-invariant*. - Normal subgroup

### Related properties

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

## Metaproperties

### Transitivity

This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property in the whole group.ABOUT THIS PROPERTY: View variations of this property that are transitive | View variations of this property that are not transitiveABOUT TRANSITIVITY: View a complete list of transitive subgroup properties|View a complete list of facts related to transitivity of subgroup properties |Read a survey article on proving transitivity

The property of being injective endomorphism-invariant is transitive on account of its being a balanced subgroup property (function restriction formalism).

`For full proof, refer: Injective endomorphism-invariance is transitive`

`Further information: Balanced implies transitive, full invariance is transitive, characteristicity is transitive`

### Trimness

This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself).

View other trim subgroup properties | View other trivially true subgroup properties | View other identity-true subgroup properties

The trivial subgroup is injective endomorphism-invariant because every endomorphism (injective or not) must take it to itself.

The whole group is also clearly injective endomorphism-invariant.

### Intersection-closedness

YES:This subgroup property is intersection-closed: an arbitrary (nonempty) intersection of subgroups with this property, also has this property.ABOUT THIS PROPERTY: View variations of this property that are intersection-closed | View variations of this property that are not intersection-closedABOUT INTERSECTION-CLOSEDNESS: View all intersection-closed subgroup properties (or, strongly intersection-closed properties) | View all subgroup properties that are not intersection-closed | Read a survey article on proving intersection-closedness | Read a survey article on disproving intersection-closedness

An arbitrary intersection of injective endomorphism-invariant subgroups is injective endomorphism-invariant. This follows on account of injective endomorphism-invariance being an invariance property.

`For full proof, refer: Injective endomorphism-invariance is strongly intersection-closed`

`Further information: Invariance implies strongly intersection-closed, normality is strongly intersection-closed, characteristicity is strongly join-closed, full invariance is strongly join-closed`

### Join-closedness

YES:This subgroup property is join-closed: an arbitrary (nonempty) join of subgroups with this property, also has this property.ABOUT THIS PROPERTY: View variations of this property that are join-closed | View variations of this property that are not join-closedABOUT JOIN-CLOSEDNESS: View all join-closed subgroup properties (or, strongly join-closed properties) | View all subgroup properties that are not join-closed | Read a survey article on proving join-closedness | Read a survey article on disproving join-closedness

An arbitrary join of injective endomorphism-invariant subgroups is injective endomorphism-invariant. This follows on account of injective endomorphism-invariance being an endo-invariance property.

`For full proof, refer: Injective endomorphism-invariance is strongly join-closed`

`Further information: Endo-invariance implies strongly join-closed, normality is strongly join-closed, characteristicity is strongly join-closed, full invariance is strongly join-closed`