Injective endomorphism-invariant subgroup

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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

Symbol-free definition

A subgroup of a group is termed injective endomorphism-invariant if every injective endomorphism of the whole group takes the subgroup to within itself.

Definition with symbols

A subgroup H of a group G is termed injective endomorphism-invariant if for any injective endomorphism \sigma of G, the image of H under \sigma is contained inside H.

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

Related properties

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