Injective endomorphism-invariant subgroup

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
For its use outside the wiki, please define the term when using it. If you are aware of an equivalent standard term, please leave a comment on the talk page
VIEW: Definitions built on this | Facts about this: (facts closely related to Injective endomorphism-invariant subgroup, all facts related to Injective endomorphism-invariant subgroup) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
Learn more about terminology local to the wiki | view a complete list of such terminology

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]
|Get subgroup property lookup help |Get exploration suggestions
VIEW RELATED: Subgroup property implications | Subgroup property non-implications | Subgroup metaproperty satisfactions | Subgroup metaproperty dissatisfactions | Subgroup property satisfactions |Subgroup property dissatisfactions
VIEW FACTS THAT:use, or start with, a subgroup satisfying the property|prove, or show that, a subgroup satisfies the property

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 $H$ of a group $G$ is called a characteristic subgroup of $G$ if ...
1	injective endomorphism-invariant	every injective endomorphism of the whole group takes the subgroup to within itself	for every injective endomorphism $φ$ of $G$ , $φ (H) \subseteq H$ . More explicitly, for any $h \in H$ and $φ \in E n d (G)$ that is injective, $φ (h) \in H$
2	injective endomorphisms restrict to endomorphisms	every injective endomorphism of the group restricts to an endomorphism of the subgroup.	for every injective endomorphism $φ$ of $G$ , $φ (H) \subseteq H$ and $φ$ restricts to an endomorphism of $H$
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 $φ$ of $G$ , $φ (H) = H$ and $φ$ restricts to an injective endomorphism of $H$

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	$H$ is an injective endomorphism-invariant subgroup of $G$ if ...	This means that injective endomorphism-invariance is ...	Additional comments
injective endomorphism $\to$ function	every injective endomorphism of $G$ sends every element of $H$ to within $H$	the invariance property for injective endomorphisms
injective endomorphism $\to$ endomorphism	every injective endomorphism of $G$ restricts to an endomorphism of $H$	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 $\to$ injective endomorphism	every injective endomorphism of $G$ restricts to a injective endomorphism of $H$	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

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 transitive
ABOUT 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-closed
ABOUT 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-closed
ABOUT 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