Difference between revisions of "Injective endomorphism-invariant subgroup"

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(Metaproperties)
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==Metaproperties==
 
==Metaproperties==
  
{{transitive}}
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The property of being injective endomorphism-invariant is transitive on account of its being a {{balanced subgroup property}}.
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Here is a summary:
  
{{proofat|[[Injective endomorphism-invariance is transitive]]}}
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{| class="sortable" border="1"
 
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!Metaproperty name !! Satisfied? !! Proof !! Difficulty level (0-5) !! Statement with symbols
{{further|[[Balanced implies transitive]], [[full invariance is transitive]], [[characteristicity is transitive]]}}
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|[[satisfies metaproperty::transitive subgroup property]] || Yes || [[injective endomorphism-invariance is transitive]] || {{#show: injective endomorphism-invariance is transitive | ?Difficulty level}} || If <math>H \le K \le G</math> are groups such that <math>H</math> is injective endomorphism-invariant in <math>K</math> and <math>K</math> is injective endomorphism-invariant in <math>G</math>, then <math>H</math> is injective endomorphism-invariant in <math>G</math>.
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| [[satisfies metaproperty::trim subgroup property]] || Yes || Obvious reasons || 0 || In any group <math>G</math>, the trivial subgroup <math>\{ e \}</math> and the whole group <math>G</math> are injective endomorphism-invariant in <math>G</math>
The trivial subgroup is injective endomorphism-invariant because every endomorphism (injective or not) must take it to itself.
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|[[satisfies metaproperty::strongly intersection-closed subgroup property]] || Yes || [[injective endomorphism-invariance is strongly intersection-closed]] || {{#show: injective endomorphism-invariance is strongly intersection-closed| ?Difficulty level}}|| If <math>H_i, i \in I</math>, are all injective endomorphism-invariant in <math>G</math>, so is the [[intersection of subgroups]] <math>\bigcap_{i \in I} H_i</math>.
The whole group is also clearly injective endomorphism-invariant.
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|[[satisfies metaproperty::strongly join-closed subgroup property]] || Yes || [[injective endomorphism-invariance is strongly join-closed]] || {{#show: injective endomorphism-invariance is strongly join-closed| ?Difficulty level}} || If <math>H_i, i \in I</math>, are all injective endomorphism-invariant in <math>G</math>, so is the [[join of subgroups]] <math>\langle H_i \rangle_{i \in I}</math>
{{intersection-closed}}
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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|[[Injective endomorphism-invariance is strongly intersection-closed]]}}
 
 
 
{{further|[[Invariance implies strongly intersection-closed]], [[normality is strongly intersection-closed]], [[characteristicity is strongly join-closed]], [[full invariance is strongly join-closed]]}}
 
 
 
{{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|[[Injective endomorphism-invariance is strongly join-closed]]}}
 
 
 
{{further|[[Endo-invariance implies strongly join-closed]], [[normality is strongly join-closed]], [[characteristicity is strongly join-closed]], [[full invariance is strongly join-closed]]}}
 

Revision as of 01:03, 18 March 2019

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


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 H of a group G is called a characteristic subgroup of G if ...
1 injective endomorphism-invariant every injective endomorphism of the whole group takes the subgroup to within itself for every injective endomorphism \varphi of G, \varphi(H) \subseteq H. More explicitly, for any h \in H and \varphi \in \operatorname{End}(G) that is injective, \varphi(h) \in H
2 injective endomorphisms restrict to endomorphisms every injective endomorphism of the group restricts to an endomorphism of the subgroup. for every injective endomorphism \varphi of G, \varphi(H) \subseteq H and \varphi restricts to an endomorphism of H
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 \varphi of G, \varphi(H) = H and \varphi restricts to an injective endomorphism of H
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
Function restriction expression H is an injective endomorphism-invariant subgroup of G if ... This means that injective endomorphism-invariance is ... Additional comments
injective endomorphism \to function every injective endomorphism of G sends every element of H to within H the invariance property for injective endomorphisms
injective endomorphism \to endomorphism every injective endomorphism of G restricts to an endomorphism of H 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 \to injective endomorphism every injective endomorphism of G restricts to a injective endomorphism of H the balanced subgroup property for injective endomorphisms Hence, it is a t.i. subgroup property, both transitive and identity-true

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
fully invariant subgroup |FULL LIST, MORE INFO
isomorph-free subgroup Intermediately injective endomorphism-invariant subgroup, Isomorph-containing subgroup|FULL LIST, MORE INFO
isomorph-containing subgroup Intermediately injective endomorphism-invariant subgroup|FULL LIST, MORE INFO
intermediately injective endomorphism-invariant 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 injective endomorphism-invariant implies characteristic characteristic not implies injective endomorphism-invariant |FULL LIST, MORE INFO
normal subgroup invariant under all inner automorphisms (via characteristic) (via characteristic) Characteristic subgroup|FULL LIST, MORE INFO

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:

Metaproperty name Satisfied? Proof Difficulty level (0-5) Statement with symbols
transitive subgroup property Yes injective endomorphism-invariance is transitive If H \le K \le G are groups such that H is injective endomorphism-invariant in K and K is injective endomorphism-invariant in G, then H is injective endomorphism-invariant in G.
trim subgroup property Yes Obvious reasons 0 In any group G, the trivial subgroup \{ e \} and the whole group G are injective endomorphism-invariant in G
strongly intersection-closed subgroup property Yes injective endomorphism-invariance is strongly intersection-closed If H_i, i \in I, are all injective endomorphism-invariant in G, so is the intersection of subgroups \bigcap_{i \in I} H_i.
strongly join-closed subgroup property Yes injective endomorphism-invariance is strongly join-closed If H_i, i \in I, are all injective endomorphism-invariant in G, so is the join of subgroups \langle H_i \rangle_{i \in I}