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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]
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
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 of a group is called a characteristic subgroup of if ...
|
1 |
injective endomorphism-invariant |
every injective endomorphism of the whole group takes the subgroup to within itself |
for every injective endomorphism of , . More explicitly, for any and that is injective,
|
2 |
injective endomorphisms restrict to endomorphisms |
every injective endomorphism of the group restricts to an endomorphism of the subgroup. |
for every injective endomorphism of , and restricts to an endomorphism of
|
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 of , and restricts to an injective endomorphism of
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This definition is presented using a tabular format. |View all pages with definitions in tabular format
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.
Find other function restriction-expressible subgroup properties | View the function restriction formalism chart for a graphic placement of this property
Relation with other properties
Stronger properties
Weaker properties
Related properties
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: