# 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  of a group  is termed a strictly characteristic subgroup of  if ...
1 surjective endomorphism-invariant it is invariant under all surjective endomorphisms of the whole group. for any surjective endomorphism  of ,  or equivalently,  for all .
2 surjective endomorphism restricts to endomorphism every surjective endomorphism of the whole group restricts to an endomorphism of the group. for any surjective endomorphism  of ,  and the restriction of  to  is an endomorphism of .

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]
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]
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  are groups such that  is strictly characteristic in  and  is strictly characteristic in . Is it necessary that  be strictly characteristic in ?
trim subgroup property (wiki-local term) Yes In any group , the trivial subgroup of  is strictly characteristic, and  itself is strictly characteristic.
strongly intersection-closed subgroup property Yes strict characteristicity is strongly intersection-closed Suppose , are all strictly characteristic subgroups of a group . Then, the intersection of subgroups  is also a strictly characteristic subgroup of .
strongly join-closed subgroup property Yes strict characteristicity is strongly join-closed Suppose  are all strictly characteristic subgroups of a group . Then, the join of subgroups  is also a strictly characteristic subgroup.
quotient-transitive subgroup property Yes strict characteristicity is quotient-transitive (generalizes to quotient-balanced implies quotient-transitive) Suppose  are groups such that  is a strictly characteristic subgroup of  and, in the quotient group , the subgroup  is strictly characteristic. Then,  is a strictly characteristic subgroup of .
intermediate subgroup condition No strict characteristicity does not satisfy intermediate subgroup condition It is possible to have groups  such that  is strictly characteristic in  but  is not strictly characteristic in . (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  ? |
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 |
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 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 |
normal subgroup invariant under all inner automorphisms (via characteristic) (via characteristic) Characteristic subgroup|FULL LIST, MORE INFO

## 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
View other second-order subgroup properties

The property of being strictly characteristic is second-order. A subgroup  is strictly characteristic in a group  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  is a strictly characteristic subgroup of  if ... This means that strict characteristicity is ... Additional comments
surjective endomorphism  function every surjective endomorphism of  sends every element of  to within  the invariance property for surjective endomorphisms
surjective endomorphism  endomorphism every surjective endomorphism of  restricts to an endomorphism of  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.