Closure-characteristic subgroup: Difference between revisions

Latest revision as of 17:30, 21 December 2014

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Definition

QUICK PHRASES: normal closure is characteristic, join of all conjugates is characteristic

A subgroup of a group is termed closure-characteristic, or a subgroup whose normal closure is characteristic, if its normal closure in the whole group is a characteristic subgroup. In symbols, a subgroup $H$ of a group $G$ is termed closure-characteristic if the normal closure $H^{G}$ of $H$ in $G$ is characteristic in $G$ .

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Intermediate notions
characteristic subgroup	invariant under all automorphisms		\|FULL LIST, MORE INFO
automorph-dominating subgroup	all automorphic subgroups are contained in conjugate subgroups		\|FULL LIST, MORE INFO
automorph-conjugate subgroup	all automorphic subgroups are conjugate	(via automorph-dominating)	\|FULL LIST, MORE INFO
join of automorph-conjugate subgroups	join of automorph-conjugate subgroups		\|FULL LIST, MORE INFO
Sylow subgroup	$p$ -subgroup of finite group with index relatively prime to $p$		\|FULL LIST, MORE INFO
join of Sylow subgroups	join of Sylow subgroups		\|FULL LIST, MORE INFO
Hall subgroup	subgroup of finite group whose order and index are relatively prime		\|FULL LIST, MORE INFO
contranormal subgroup	normal closure is whole group		\|FULL LIST, MORE INFO

Conjunction with other properties

Any normal subgroup that is also closure-characteristic, is characteristic.

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)

Metaproperty name	Satisfied?	Proof	Difficulty level (0-5)	Statement with symbols
trim subgroup property	Yes	(obvious)	0	In any group $G$ , both the trivial subgroup ${e}$ and the whole group $G$ are closure-characteristic.
strongly join-closed subgroup property	Yes	closure-characteristicity is strongly join-closed		Suppose $G$ is a group and $H_{i}, i \in I$ are all closure-characteristic subgroups of $G$ . Then the join $⟨ H_{i} ⟩_{i \in I}$ is also closure-characteristic.

@@ Line 5: / Line 5: @@
 ==Definition==
-===Symbol-free definition===
+{{quick phrase|[[quick phrase::normal closure is characteristic]], [[quick phrase::join of all conjugates is characteristic]]}}
-A [[subgroup]] of a [[group]] is termed '''closure-characteristic''' if its [[normal closure]] in the whole group is a [[characteristic subgroup]].
+A [[subgroup]] of a [[group]] is termed '''closure-characteristic''', or a '''subgroup whose normal closure is characteristic''', if its [[defining ingredient::normal closure]] in the whole group is a [[defining ingredient::characteristic subgroup]]. In symbols, a [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed '''closure-characteristic''' if the [[normal closure]] <math>H^G</math> of <math>H</math> in <math>G</math> is [[characteristic subgroup|characteristic]] in <math>G</math>.
-===Definition with symbols===
+==Relation with other properties==
-A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed '''closure-characteristic''' if the [[normal closure]] <math>H^G</math> of <math>H</math> in <math>G</math> is [[characteristic subgroup|characteristic]] in <math>G</math>.
+===Stronger properties===
-==Relation with other properties==
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Weaker than::characteristic subgroup]] || invariant under all [[automorphism]]s || || || {{intermediate notions short|closure-characteristic subgroup|characteristic subgroup}}
+|-
+| [[Weaker than::automorph-dominating subgroup]] || all [[automorphic subgroups]] are contained in [[conjugate subgroups]] || || || {{intermediate notions short|closure-characteristic subgroup|automorph-dominating subgroup}}
+|-
+| [[Weaker than::automorph-conjugate subgroup]] || all [[automorphic subgroups]] are [[conjugate subgroups|conjugate]] || (via automorph-dominating) || || {{intermediate notions short|closure-characteristic subgroup|automorph-conjugate subgroup}}
+|-
+| [[Weaker than::join of automorph-conjugate subgroups]] || [[join of subgroups|join]] of [[automorph-conjugate subgroup]]s || || || {{intermediate notions short|closure-characteristic subgroup|join of automorph-conjugate subgroups}}
+|-
+| [[Weaker than::Sylow subgroup]] || <math>p</math>-subgroup of finite group with index relatively prime to <math>p</math> || || || {{intermediate notions short|closure-characteristic subgroup|Sylow subgroup}}
+|-
+| [[Weaker than::join of Sylow subgroups]] || join of [[Sylow subgroup]]s || || || {{intermediate notions short|closure-characteristic subgroup|join of Sylow subgroups}}
+|-
+| [[Weaker than::Hall subgroup]] || subgroup of finite group whose order and index are relatively prime || || || {{intermediate notions short|closure-characteristic subgroup|Hall subgroup}}
+|-
+| [[Weaker than::contranormal subgroup]] || [[normal closure]] is whole group || || || {{intermediate notions short|closure-characteristic subgroup|contranormal subgroup}}
+|}
-===Stronger properties===
+===Conjunction with other properties===
-* [[Characteristic subgroup]]
+Any [[normal subgroup]] that is also closure-characteristic, is characteristic.[[Category:Normal-to-characteristic subgroup properties]]
-* [[Automorph-conjugate subgroup]]
-* [[Join of automorph-conjugate subgroups]]
-* [[Sylow subgroup]]
-* [[Hall subgroup]]
 ==Metaproperties==
-{{trim}}
+{{wikilocal-section}}
-{{join-closed}}
+{| class="sortable" border="1"
+!Metaproperty name !! Satisfied? !! Proof !! Difficulty level (0-5) !! Statement with symbols
+|-
+| [[satisfies metaproperty::trim subgroup property]] || Yes || (obvious) || 0 || In any group <math>G</math>, both the trivial subgroup <math>\{ e \}</math> and the whole group <math>G</math> are closure-characteristic.
+|-
+| [[satisfies metaproperty::strongly join-closed subgroup property]] || Yes || [[closure-characteristicity is strongly join-closed]] || {{#show: closure-characteristicity is strongly join-closed| ?Difficulty level}} || Suppose <math>G</math> is a group and <math>H_i, i \in I</math> are all closure-characteristic subgroups of <math>G</math>. Then the join <math>\langle H_i \rangle_{i \in I}</math> is also closure-characteristic.
+|}