Injective endomorphism-invariant subgroup: Difference between revisions

Revision as of 20:38, 15 June 2009

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]
|Get subgroup property lookup help |Get exploration suggestions
VIEW RELATED: Subgroup property implications | Subgroup property non-implications | Subgroup metaproperty satisfactions | Subgroup metaproperty dissatisfactions | Subgroup property satisfactions |Subgroup property dissatisfactions
VIEW FACTS THAT:use, or start with, a subgroup satisfying the property|prove, or show that, a subgroup satisfies the property

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

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
For its use outside the wiki, please define the term when using it. If you are aware of an equivalent standard term, please leave a comment on the talk page
VIEW: Definitions built on this | Facts about this: (facts closely related to Injective endomorphism-invariant subgroup, all facts related to Injective endomorphism-invariant subgroup) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
Learn more about terminology local to the wiki | view a complete list of such terminology

Definition

Symbol-free definition

A subgroup of a group is termed injective endomorphism-invariant or I-characteristic if every injective endomorphism of the whole group takes the subgroup to within itself.

Definition with symbols

A subgroup $H$ of a group $G$ is termed injective endomorphism-invariant or I-characteristic if for any injective endomorphism $σ$ of $G$ , the image of $H$ under $σ$ is contained inside $H$ .

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

In the function restriction formalism, the subgroup property of being strictly characteristic can be expressed in any of the following ways:

As the invariance property with respect to injective endomorphisms, namely:

Injective endomorphism $\to$ Function

As the following:

Injective endomorphism $\to$ Endomorphism

As the balanced subgroup property (function restriction formalism) with respect to injective endomorphisms:

Injective endomorphism $\to$ Injective endomorphism

Relation with other properties

Stronger properties

Weaker properties

Characteristic subgroup: For proof of the implication, refer I-characteristic implies characteristic and for proof of its strictness (i.e. the reverse implication being false) refer Characteristic not implies I-characteristic.
Normal subgroup

Related properties

Strictly characteristic subgroup: This is the invariance property with respect to surjective, rather than injective, endomorphisms.

Metaproperties

Transitivity

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).

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 injective endomorphism-invariant because every endomorphism (injective or not) must take it to itself.

The whole group is also clearly injective endomorphism-invariant.

Intersection-closedness

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: 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.
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 Template:Endo-invariance property. For full proof, refer: Endo-invariance implies strongly join-closed

@@ Line 9: / Line 9: @@
 ===Symbol-free definition===
-A [[subgroup]] of a [[group]] is termed '''I-characteristic''' or '''injective endomorphism-invariant''' if every [[injective endomorphism]] of the whole group takes the subgroup to within itself.
+A [[subgroup]] of a [[group]] is termed '''injective endomorphism-invariant''' or '''I-characteristic''' if every [[injective endomorphism]] of the whole group takes the subgroup to within itself.
 ===Definition with symbols===
-A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed '''I-characteristic''' or '''injective endomorphism-invariant''' if for any [[injective endomorphism]] <math>\sigma</math> of <math>G</math>, the image of <math>H</math> under <math>\sigma</math> is contained inside <math>H</math>.
+A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed '''injective endomorphism-invariant''' or '''I-characteristic''' if for any [[injective endomorphism]] <math>\sigma</math> of <math>G</math>, the image of <math>H</math> under <math>\sigma</math> is contained inside <math>H</math>.
 ==Formalisms==
@@ Line 39: / Line 39: @@
 * [[Weaker than::Isomorph-free subgroup]]
 * [[Weaker than::Isomorph-containing subgroup]]
+* [[Weaker than::Intermediately injective endomorphism-invariant subgroup]]
 ===Weaker properties===
@@ Line 53: / Line 54: @@
 {{transitive}}
-The property of being I-characteristic is transitive on account of ist being a {{balanced subgroup property}}.
+The property of being injective endomorphism-invariant is transitive on account of its being a {{balanced subgroup property}}.
 {{trim}}
-The trivial subgroup is strictly characteristic because every endomorphism (injective or not) must take it to itself.
+The trivial subgroup is injective endomorphism-invariant because every endomorphism (injective or not) must take it to itself.
-The whole group is also clearly strictly characteristic.
+The whole group is also clearly injective endomorphism-invariant.
 {{intersection-closed}}
-An arbitrary intersection of I-characteristic subgroups is I-characteristic. This follows on account of I-characteristicity being an {{invariance property}}. {{proofat|[[Invariance implies strongly intersection-closed]]}}
+An arbitrary intersection of injective endomorphism-invariant subgroups is injective endomorphism-invariant. This follows on account of injective endomorphism-invariance being an {{invariance property}}. {{proofat|[[Invariance implies strongly intersection-closed]]}}
 {{join-closed}}
-An arbitrary join of I-characteristic subgroups is I-characteristic. This follows on account of I-characteristicity being an {{endo-invariance property}}. {{proofat|[[Endo-invariance implies strongly join-closed]]}}
+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}}. {{proofat|[[Endo-invariance implies strongly join-closed]]}}