Injective endomorphism-invariant subgroup

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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]

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

• 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.
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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.
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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
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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