Injective endomorphism-invariant subgroup

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

Symbol-free definition

A subgroup of a group is termed I-characteristic or 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 I-characteristic or 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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In the function restriction formalism, the subgroup property of being strictly characteristic can be expressed in any of the following ways:

• As the invariance property with respect to injective endomorphisms, namely:

Injective endomorphism $\to$ Function

• As the following:

Injective endomorphism $\to$ Endomorphism

• As the balanced subgroup property (function restriction formalism) with respect to injective endomorphisms:

Injective endomorphism $\to$ Injective endomorphism

Relation with other properties

Stronger properties

• Fully characteristic subgroup
• Isomorph-free subgroup
• Isomorph-containing subgroup

Weaker properties

• Characteristic subgroup: For proof of the implication, refer I-characteristic implies characteristic and for proof of its strictness (i.e. the reverse implication being false) refer Characteristic not implies I-characteristic.
• 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 I-characteristic is transitive on account of ist being a balanced subgroup property (function restriction formalism).

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

The whole group is also clearly strictly characteristic.

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 I-characteristic subgroups is I-characteristic. This follows on account of I-characteristicity being an invariance property. For full proof, refer: Invariance implies strongly intersection-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 I-characteristic subgroups is I-characteristic. This follows on account of I-characteristicity being an Template:Endo-invariance property. For full proof, refer: Endo-invariance implies strongly join-closed