Injective endomorphism-invariant subgroup
BEWARE! This term is nonstandard and is being used locally within the wiki. [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
QUICK PHRASES: invariant under all injective endomorphisms, injective endomorphism-invariant
Definition with symbols
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 an injective endomorphism-invariant subgroup of if ...||This means that injective endomorphism-invariance is ...||Additional comments|
|injective endomorphism function||every injective endomorphism of sends every element of to within||the invariance property for injective endomorphisms|
|injective endomorphism endomorphism||every injective endomorphism of restricts to an endomorphism of||the endo-invariance property for injective endomorphisms; i.e., it is the invariance property for injective endomorphism, which is a property stronger than the property of being an endomorphism|
|injective endomorphism injective endomorphism||every injective endomorphism of restricts to a injective endomorphism of||the balanced subgroup property for injective endomorphisms||Hence, it is a t.i. subgroup property, both transitive and identity-true|
Relation with other properties
- Fully characteristic subgroup
- Isomorph-free subgroup
- Isomorph-containing subgroup
- Intermediately injective endomorphism-invariant subgroup
- Characteristic subgroup: For proof of the implication, refer injective endomorphism-invariant implies characteristic and for proof of its strictness (i.e. the reverse implication being false) refer Characteristic not implies injective endomorphism-invariant.
- Normal subgroup
- Strictly characteristic subgroup: This is the invariance property with respect to surjective, rather than injective, endomorphisms.
This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property in the whole group.
ABOUT THIS PROPERTY: View variations of this property that are transitive | View variations of this property that are not transitive
ABOUT TRANSITIVITY: View a complete list of transitive subgroup properties|View a complete list of facts related to transitivity of subgroup properties |Read a survey article on proving transitivity
The property of being injective endomorphism-invariant is transitive on account of its being a balanced subgroup property (function restriction formalism).
For full proof, refer: Injective endomorphism-invariance is transitive
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 injective endomorphism-invariant because every endomorphism (injective or not) must take it to itself.
The whole group is also clearly injective endomorphism-invariant.
YES: This subgroup property is intersection-closed: an arbitrary (nonempty) intersection of subgroups with this property, also has this property.
ABOUT THIS PROPERTY: View variations of this property that are intersection-closed | View variations of this property that are not intersection-closed
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
An arbitrary intersection of injective endomorphism-invariant subgroups is injective endomorphism-invariant. This follows on account of injective endomorphism-invariance being an invariance property.
For full proof, refer: Injective endomorphism-invariance is strongly intersection-closed
YES: This subgroup property is join-closed: an arbitrary (nonempty) join of subgroups with this property, also has this property.
ABOUT THIS PROPERTY: View variations of this property that are join-closed | View variations of this property that are not join-closed
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
An arbitrary join of injective endomorphism-invariant subgroups is injective endomorphism-invariant. This follows on account of injective endomorphism-invariance being an endo-invariance property.
For full proof, refer: Injective endomorphism-invariance is strongly join-closed