Injective endomorphism-invariant subgroup: Difference between revisions

Latest revision as of 01:04, 18 March 2019

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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]
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If the ambient group is a finite group, this property is equivalent to the property: characteristic subgroup
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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				\|FULL LIST, MORE INFO
isomorph-containing 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

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

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	follows from balanced implies transitive		If $H\leq K\leq G$ 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	follows from invariance implies 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	follows from endo-invariance implies strongly join-closed		If $H_{i},i\in I$ , are all injective endomorphism-invariant in $G$ , so is the join of subgroups $\langle H_{i}\rangle _{i\in I}$

@@ Line 7: / Line 7: @@
 {{quick phrase|[[quick phrase::invariant under all injective endomorphisms]], [[quick phrase::injective endomorphism-invariant]]}}
-===Symbol-free definition===
-A [[subgroup]] of a [[group]] is termed '''injective endomorphism-invariant''' if every [[injective endomorphism]] of the whole group takes the subgroup to within itself.
+===Equivalent definitions in tabular format===
-===Definition with symbols===
+Below are many '''equivalent''' definitions of injective endomorphism-invariant subgroup.
-A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed '''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>.
+{| class="sortable" border="1"
+! No. !! Shorthand !! A [[subgroup]] of a [[group]] is characteristic in it if... !! A subgroup <math>H</math> of a group <math>G</math> is called a characteristic subgroup of <math>G</math> if ...
+|-
+| 1 || injective endomorphism-invariant || every [[injective endomorphism]] of the whole group takes the subgroup to within itself || for every [[injective endomorphism]] <math>\varphi</math> of <math>G</math>, <math>\varphi(H) \subseteq H</math>. More explicitly, for any <math>h \in H</math> and <math>\varphi \in \operatorname{End}(G)</math> that is injective, <math>\varphi(h) \in H</math>
+|-
+| 2 || injective endomorphisms restrict to endomorphisms || every injective endomorphism of the group restricts to an [[endomorphism]] of the subgroup. || for every injective endomorphism <math>\varphi</math> of <math>G</math>, <math>\varphi(H) \subseteq H</math> and <math>\varphi</math> restricts to an [[endomorphism]] of <math>H</math>
+|-
+| 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 <math>\varphi</math> of <math>G</math>, <math>\varphi(H) = H</math> and <math>\varphi</math> restricts to an [[injective endomorphism]] of <math>H</math>
+|}
+{{tabular definition format}}
 ==Formalisms==
@@ Line 32: / Line 40: @@
 ===Stronger properties===
-* [[Weaker than::Fully characteristic subgroup]]
+{| class="sortable" border="1"
-* [[Weaker than::Isomorph-free subgroup]]
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
-* [[Weaker than::Isomorph-containing subgroup]]
+|-
-* [[Weaker than::Intermediately injective endomorphism-invariant subgroup]]
+|  [[Weaker than::fully invariant subgroup]] || || || || {{intermediate notions short|injective endomorphism-invariant subgroup|fully invariant subgroup}}
+|-
+| [[Weaker than::isomorph-free subgroup]] || || || || {{intermediate notions short|injective endomorphism-invariant subgroup|isomorph-free subgroup}}
+|-
+| [[Weaker than::isomorph-containing subgroup]] || || || || {{intermediate notions short|injective endomorphism-invariant subgroup|isomorph-containing subgroup}}
+|-
+| [[Weaker than::intermediately injective endomorphism-invariant subgroup]] || || || || {{intermediate notions short|injective endomorphism-invariant subgroup|intermediately injective endomorphism-invariant subgroup}}
+|}
 ===Weaker properties===
-* [[Stronger than::Characteristic subgroup]]: {{proofofstrictimplicationat|[[injective endomorphism-invariant implies characteristic]]|[[Characteristic not implies injective endomorphism-invariant]]}}
+{| class="sortable" border="1"
-* [[Stronger than::Normal subgroup]]
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+|  [[Stronger than::characteristic subgroup]] || invariant under all automorphisms || [[injective endomorphism-invariant implies characteristic]] || [[characteristic not implies injective endomorphism-invariant]] || {{intermediate notions short|characteristic subgroup|injective endomorphism-invariant subgroup}}
+|-
+| [[Stronger than::normal subgroup]] || invariant under all inner automorphisms || (via characteristic) || (via characteristic) || {{intermediate notions short|normal subgroup|injective endomorphism-invariant subgroup}}
+|}
 ===Related properties===
@@ Line 48: / Line 68: @@
 ==Metaproperties==
-{{transitive}}
+{{wikilocal-section}}
-The property of being injective endomorphism-invariant is transitive on account of its being a {{balanced subgroup property}}.
+Here is a summary:
-{{proofat|[[Injective endomorphism-invariance is transitive]]}}
+{| class="sortable" border="1"
+!Metaproperty name !! Satisfied? !! Proof !! Difficulty level (0-5) !! Statement with symbols
-{{further|[[Balanced implies transitive]], [[full invariance is transitive]], [[characteristicity is transitive]]}}
+|-
+|[[satisfies metaproperty::transitive subgroup property]] || Yes || follows from [[balanced implies transitive]] || || 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>.
-{{trim}}
+|-
+| [[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.
+|-
+|[[satisfies metaproperty::strongly intersection-closed subgroup property]] || Yes || follows from [[invariance implies 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.
+|-
+|[[satisfies metaproperty::strongly join-closed subgroup property]] || Yes || follows from [[endo-invariance implies 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}}
+|}
-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]]}}