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Contents 1 History 1.1 Origin of the concept 2 Definition 2.1 Symbol-free definition 2.2 Definition with symbols 3 Formalisms 3.1 Second-order description 3.2 Function restriction expression 4 Relation with other properties 4.1 Stronger properties 4.2 Weaker properties 4.3 Related properties 5 Metaproperties 5.1 Transitivity 5.2 Trimness 5.3 Intersection-closedness 5.4 Join-closedness 5.5 Quotient-transitivity 6 Effect of property operators 6.1 Left transiter 6.2 The left transiter 6.3 Right transiter 6.4 The right transiter 6.5 Subordination 7 References 7.1 Journal references

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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View other semistandard definitions in group theory

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof.
View a complete list of subgroup properties|Get subgroup property lookup help |Get exploration suggestions[SHOW MORE]

VIEW RELATED: Subgroup property implications | Subgroup property non-implications | Subgroup metaproperty satisfactions |
VIEW FACTS THAT: use, or start with, a subgroup satisfying the property| prove, or show that, a subgroup satisfies the property
RANDOM SUBGROUP PROPERTY: Hall subgroup: A subgroup of a finite group whose order and index are relatively prime. Sylow subgroups are a special case.

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 characteristicity
Find other variations of characteristicity | Read a survey article on varying characteristicity

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.

Definition

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

Symbol-free definition

A subgroup of a group is termed strictly characteristic (or distinguished or surjective endomorphism-invariant) if any surjective endomorphism of the whole group takes the subgroup to within itself. In other words, any surjective endomorphism of the whole group must restrict to an endomorphism of the subgroup.

Definition with symbols

A subgroup H of a group G is termed strictly characteristic (or distinguished or surjective endomorphism-invariant) if for any surjective endomorphism σ of G, the image of H under σ is contained inside H.

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
View other second-order subgroup properties

The property of being strictly characteristic is second-order. A subgroup H is strictly characteristic in a group G if:

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 function	every surjective endomorphism of G sends every element of H to within H	the invariance property for surjective endomorphisms
surjective endomorphism 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

Relation with other properties

Stronger properties

property	quick description	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)	click here
Completely strictly characteristic subgroup (or completely distinguished subgroup)	equals its own pre-image under any surjective endomorphism			click here
Surjective endomorphism-balanced subgroup	surjective endomorphism of whole group restricts to surjective endomorphism of subgroup
Intermediately strictly characteristic subgroup	strictly characteristic in every intermediate subgroup
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	click here
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	click here
* Prehomomorph-contained subgroup	contained in any subgroup having it as a homomorphic image	prehomomorph-contained implies strictly characteristic	strictly characteristic not implies prehomomorph-contained	click here

Weaker properties

property	quick description	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
Normal subgroup	invariant under all inner automorphisms	(via characteristic)	(via characteristic)	click here

Related properties

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

Metaproperties

Transitivity

NO: This subgroup property is not transitive: a subgroup with this property in a subgroup with this property, need not have the property in the whole group
ABOUT THIS PROPERTY: |
ABOUT TRANSITIVITY: View a complete list of subgroup properties that are not transitive| View facts related to transitivity of subgroup properties | View a survey article on disproving transitivity

The property of being strictly characteristic is not transitive. That is, the following situation is possible: H is a strictly characteristic subgroup of K, K is a strictly characteristic subgroup of G, but H is not a strictly characteristic subgroup of G.

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 strictly characteristic because every endomorphism (surjective 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.
In fact, since the property is also true for every group as a subgroup of itself, it is a strongly intersection-closed subgroup property.
ABOUT THIS PROPERTY: |
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

On account of its being an invariance property, the subgroup property of being strictly characteristic is closed under arbitrary intersections.

For full proof, refer: Strict characteristicity is strongly intersection-closed

Further information: 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.
In fact, since the property is also true for the trivial subgroup in any group, it is a strongly join-closed subgroup property.
ABOUT THIS PROPERTY: |
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

The subgroup generated by an arbitrary family of strictly characteristic subgroups is strictly characteristic. This is on account of its being an endo-invariance property.

For full proof, refer: Strict characteristicity is strongly join-closed

Further information: Endo-invariance implies strongly join-closed

Quotient-transitivity

This subgroup property is quotient-transitive: the corresponding quotient property is transitive.
View a complete list of quotient-transitive subgroup properties

If are groups such that H is a strictly characteristic subgroup of G and K / H is a strictly characteristic subgroup of G / H. This follows because the property of being a strictly characteristic subgroup is a quotient-balanced subgroup property.

For full proof, refer: Strict characteristicity is quotient-transitive

Further information: Quotient-balanced implies quotient-transitive

Effect of property operators

Left transiter

The left transiter

Applying the left transiter to this property gives: left-transitively strictly characteristic subgroup

From the restriction formal expressions, the following is clear: any fully invariant subgroup of a strictly characteristic subgroup is strictly characteristic. Thus, any fully invariant subgroup is left-transitively strictly characteristic.

For full proof, refer: Fully invariant of strictly characteristic implies strictly characteristic

Right transiter

The right transiter

Applying the right transiter to this property gives: right-transitively strictly characteristic subgroup

Any strictly characteristic subgroup of a surjective endomorphism-balanced subgroup is strictly characteristic.

An even stronger property is the property of being a completely strictly characteristic subgroup: this is a subgroup that equals its full inverse image for any surjective endomorphism.

For full proof, refer: Strictly characteristic of surjective endomorphism-balanced implies strictly characteristic

Subordination

Every strictly characteristic subgroup is characteristic, and we know that the property of being characteristic is a t.i. subgroup property. Hence, in particular, the property obtained by applying the subordination operator to the subgroup property of being strictly characteristic, must at any rate be stronger than the property of being 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 characteristic subgroup (here, called fully invariant subgroup).^{Full text (PDF)}

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