Strictly characteristic subgroup

Definition

QUICK PHRASES: invariant under all surjective endomorphisms, surjective endomorphism-invariant

A subgroup of a group is termed strictly characteristic or distingushed if it satisfies the following equivalent conditions:

No.	Shorthand	A subgroup of a group is termed strictly characteristic if ...	A subgroup $H$ of a group $G$ is termed a strictly characteristic subgroup of $G$ if ...
1	surjective endomorphism-invariant	it is invariant under all surjective endomorphisms of the whole group.	for any surjective endomorphism $\varphi$ of $G$, $\varphi(H) \subseteq H$ or equivalently, $\varphi(x) \in H$ for all $x \in H$.
2	surjective endomorphism restricts to endomorphism	every surjective endomorphism of the whole group restricts to an endomorphism of the group.	for any surjective endomorphism $\varphi$ of $G$, $\varphi(H) \subseteq H$ and the restriction of $\varphi$ to $H$ is an endomorphism of $H$.

This article defines a term that has been used or referenced in a journal article or standard publication, but may not be generally accepted by the mathematical community as a standard term.[SHOW MORE]

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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
View other properties finitarily equivalent to characteristic subgroup | View other variations of characteristic subgroup | Read a survey article on varying characteristic subgroup

This is a variation of characteristic subgroup|Find other variations of characteristic subgroup | Read a survey article on varying characteristic subgroup

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
transitive subgroup property	unclear	--	Suppose $H \le K \le G$ are groups such that $H$ is strictly characteristic in $K$ and $K$ is strictly characteristic in $G$. Is it necessary that $H$ be strictly characteristic in $G$?
trim subgroup property (wiki-local term)	Yes		In any group $G$, the trivial subgroup of $G$ is strictly characteristic, and $G$ itself is strictly characteristic.
strongly intersection-closed subgroup property	Yes	strict characteristicity is strongly intersection-closed	Suppose $H_i, i \in I$, are all strictly characteristic subgroups of a group $G$. Then, the intersection of subgroups $\bigcap_{i \in I} H_i$ is also a strictly characteristic subgroup of $G$.
strongly join-closed subgroup property	Yes	strict characteristicity is strongly join-closed	Suppose $H_i, i \in I$ are all strictly characteristic subgroups of a group $G$. Then, the join of subgroups $\langle H_i \rangle_{i \in I}$ is also a strictly characteristic subgroup.
quotient-transitive subgroup property	Yes	strict characteristicity is quotient-transitive (generalizes to quotient-balanced implies quotient-transitive)	Suppose $H \le K \le G$ are groups such that $H$ is a strictly characteristic subgroup of $G$ and, in the quotient group $G/H$, the subgroup $K/H$ is strictly characteristic. Then, $K$ is a strictly characteristic subgroup of $G$.
intermediate subgroup condition	No	strict characteristicity does not satisfy intermediate subgroup condition	It is possible to have groups $H \le K \le G$ such that $H$ is strictly characteristic in $G$ but $H$ is not strictly characteristic in $K$. (In fact, we can choose any of the finite examples for characteristicity does not satisfy intermediate subgroup condition).

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
fully invariant subgroup	invariant under all endomorphisms		strictly characteristic not implies fully invariant (see also list of examples)	Normality-preserving endomorphism-invariant subgroup|FULL LIST, MORE INFO
bound-word subgroup	governed by words and equations	bound-word implies strictly characteristic	?	|FULL LIST, MORE INFO
marginal subgroup		(via bound-word)	?	Bound-word subgroup, Weakly marginal subgroup|FULL LIST, MORE INFO
completely strictly characteristic subgroup (or completely distinguished subgroup)	equals its own pre-image under any surjective endomorphism			Surjective endomorphism-balanced subgroup|FULL LIST, MORE INFO
surjective endomorphism-balanced subgroup	surjective endomorphism of whole group restricts to surjective endomorphism of subgroup			|FULL LIST, MORE INFO
intermediately strictly characteristic subgroup	strictly characteristic in every intermediate subgroup			|FULL LIST, MORE INFO
normal-homomorph-containing subgroup	contains any normal subgroup that is a homomorphic image of it	normal-homomorph-containing implies strictly characteristic	strictly characteristic not implies normal-homomorph-containing	Normality-preserving endomorphism-invariant subgroup, Weakly normal-homomorph-containing subgroup|FULL LIST, MORE INFO
weakly normal-homomorph-containing subgroup	contains any homomorphic image in a map that sends its normal subgroups ot normal subgroups	weakly normal-homomorph-containing implies strictly characteristic	strictly characteristic not implies weakly normal-homomorph-containing	Normality-preserving endomorphism-invariant subgroup|FULL LIST, MORE INFO
Prehomomorph-contained subgroup	contained in any subgroup having it as a homomorphic image	prehomomorph-contained implies strictly characteristic	strictly characteristic not implies prehomomorph-contained	Intermediately strictly characteristic 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	strictly characteristic implies characteristic	characteristic not implies strictly characteristic	|FULL LIST, MORE INFO
normal subgroup	invariant under all inner automorphisms	(via characteristic)	(via characteristic)	Characteristic subgroup|FULL LIST, MORE INFO

Related properties

• Injective endomorphism-invariant subgroup
• Retraction-invariant subgroup, retraction-invariant normal subgroup, retraction-invariant characteristic subgroup

Effect of property operators

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)

Operator	Meaning	Result of application	Proof and additional observations
left-transitively operator	if big group is strictly characteristic in a bigger group, so is subgroup	left-transitively strictly characteristic subgroup	by definition; also note that any fully invariant subgroup satisfies the property.
right-transitively operator	every strictly characteristic subgroup of the subgroup is strictly characteristic in the whole group	right-transitively strictly characteristic subgroup	by definition; also note that any surjective endomorphism-balanced subgroup satisfies the property.
subordination operator	strictly characteristic subgroup of strictly characteristic subgroup of ... strictly characteristic subgroup	sub-strictly characteristic subgroup	stronger than characteristic subgroup

Formalisms

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)

Second-order description

This subgroup property is a second-order subgroup property, viz., it has a second-order description in the theory of groups
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The property of being strictly characteristic is second-order. A subgroup $H$ is strictly characteristic in a group $G$ if:

$\ \forall g \in H, \sigma \in G^G, (\ \forall \ a,b \in G, \sigma(ab) = \sigma(a)\sigma(b)) \land (\ \forall a \in G, \exists b: \sigma(b) = a) \implies \sigma(g) \in H$

Note that the two conditions checked parenthetically are respectively the conditions of being an endomorphism and being surjective.

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 a strictly characteristic subgroup of $G$ if ...	This means that strict characteristicity is ...	Additional comments
surjective endomorphism $\to$ function	every surjective endomorphism of $G$ sends every element of $H$ to within $H$	the invariance property for surjective endomorphisms
surjective endomorphism $\to$ endomorphism	every surjective endomorphism of $G$ restricts to an endomorphism of $H$	the endo-invariance property for surjective endomorphisms; i.e., it is the invariance property for surjective endomorphism, which is a property stronger than the property of being an endomorphism

History

Origin of the concept

The concept has been explored under two names: strictly characteristic subgroup and distinguished subgroup. The first term has been used in Bourbaki's texts in a more general context of algebras.

The term strictly characteristic was used by Reinhold Baer in his paper The Higher Commutator Subgroups of a Group where he compares invariance properties like being normal, characteristic, strictly characteristic and fully characteristic.

References

Journal references

• The higher commutator subgroups of a group by Reinhold Baer, Bulletin of the American Mathematical Society, ISSN 10889485 (electronic), ISSN 02730979 (print), Page 143 - 160(Year 1944): This paper compares invariance properties such as normal subgroup, characteristic subgroup, strictly characteristic subgroup, and fully invariant subgroup.^{Full text (PDF)}

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