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
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
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.
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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