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

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 $\varphi$ of $G$, $\varphi(H) \subseteq H$. More explicitly, for any $h \in H$ and $\varphi \in \operatorname{End}(G)$ that is injective, $\varphi(h) \in H$
2	injective endomorphisms restrict to endomorphisms	every injective endomorphism of the group restricts to an endomorphism of the subgroup.	for every injective endomorphism $\varphi$ of $G$, $\varphi(H) \subseteq H$ and $\varphi$ 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 $\varphi$ of $G$, $\varphi(H) = H$ and $\varphi$ restricts to an injective endomorphism of $H$

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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

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
fully invariant subgroup				|FULL LIST, MORE INFO
isomorph-free subgroup				Intermediately injective endomorphism-invariant subgroup, Isomorph-containing subgroup|FULL LIST, MORE INFO
isomorph-containing subgroup				Intermediately injective endomorphism-invariant subgroup|FULL LIST, MORE INFO
intermediately injective endomorphism-invariant subgroup				|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
characteristic subgroup	invariant under all automorphisms	injective endomorphism-invariant implies characteristic	characteristic not implies injective endomorphism-invariant	|FULL LIST, MORE INFO
normal subgroup	invariant under all inner automorphisms	(via characteristic)	(via characteristic)	Characteristic subgroup|FULL LIST, MORE INFO

Related properties

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

Metaproperties

BEWARE! This section of the article uses terminology local to the wiki, possibly without giving a full explanation of the terminology used (though efforts have been made to clarify terminology as much as possible within the particular context)

Here is a summary:

Metaproperty name	Satisfied?	Proof	Difficulty level (0-5)	Statement with symbols
transitive subgroup property	Yes	follows from balanced implies transitive		If $H \le K \le G$ are groups such that $H$ is injective endomorphism-invariant in $K$ and $K$ is injective endomorphism-invariant in $G$, then $H$ is injective endomorphism-invariant in $G$.
trim subgroup property	Yes	Obvious reasons	0	In any group $G$, the trivial subgroup $\{ e \}$ and the whole group $G$ are injective endomorphism-invariant in $G$
strongly intersection-closed subgroup property	Yes	follows from invariance implies strongly intersection-closed		If $H_i, i \in I$, are all injective endomorphism-invariant in $G$, so is the intersection of subgroups $\bigcap_{i \in I} H_i$.
strongly join-closed subgroup property	Yes	follows from endo-invariance implies strongly join-closed		If $H_i, i \in I$, are all injective endomorphism-invariant in $G$, so is the join of subgroups $\langle H_i \rangle_{i \in I}$