Transfer-closed characteristic subgroup: Difference between revisions

Latest revision as of 23:54, 31 May 2020

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Definition

Definition with symbols

A subgroup $H$ of a group $G$ is termed a transfer-closed characteristic subgroup if, for any subgroup $K\leq G$ , $H\cap K$ is a characteristic subgroup of $K$ .

Formalisms

In terms of the transfer condition operator

This property is obtained by applying the transfer condition operator to the property: characteristic subgroup
View other properties obtained by applying the transfer condition operator

Relation with other properties

Stronger properties

Weaker properties

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
transitive subgroup property	Yes	transfer-closed characteristicity is transitive	If $H\leq K\leq G$ are groups such that $H$ is transfer-closed characteristic in $K$ and $K$ is transfer-closed characteristic in $G$ , then $H$ is transfer-closed characteristic in $G$ .
intermediate subgroup condition	Yes		If $H\leq K\leq G$ are groups such that $H$ is transfer-closed characteristic in $G$ , then $H$ is transfer-closed characteristic in $K$ .
strongly intersection-closed subgroup property	Yes		If $H_{i},i\in I$ is a family of transfer-closed characteristic subgroups of a group $G$ , the intersection $\bigcap _{i\in I}H_{i}$ is also a transfer-closed characteristic subgroup of $G$ .
transfer condition	Yes		If $H$ and $K$ are subgroups of $G$ such that $H$ is transfer-closed characteristic in $G$ , then $H\cap K$ is transfer-closed characteristic in $K$ .

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 {{wikilocal}}
 {{subgroup property}}
+{{variation of|characteristicity}}
 ==Definition==
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 ==Formalisms==
-{{obtainedbyapplyingthe|transfer-transiter|characteristic subgroup}}
+{{obtainedbyapplyingthe|transfer condition operator|characteristic subgroup}}
 ==Relation with other properties==
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 ===Stronger properties===
+* [[Weaker than::Normal Sylow subgroup]]
 * [[Weaker than::Normal Hall subgroup]]
+* [[Weaker than::Variety-containing subgroup]]
+* [[Weaker than::Subisomorph-containing subgroup]]
 ===Weaker properties===
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 ==Metaproperties==
-{{transitive}}
+{| class="sortable" border="1"
+! Metaproperty name !! Satisfied? !! Proof !! Difficulty level !! Statement with symbols
-{{trim}}
+|-
+| [[satisfies metaproperty::transitive subgroup property]] || Yes || [[transfer-closed characteristicity is transitive]] || {{#show:transfer-closed characteristicity is transitive| ?Difficulty level}} || If <math>H \le K \le G</math> are groups such that <math>H</math> is transfer-closed characteristic in <math>K</math> and <math>K</math> is transfer-closed characteristic in <math>G</math>, then <math>H</math> is transfer-closed characteristic in <math>G</math>.
-{{intsubcondn}}
+|-
+| [[satisfies metaproperty::intermediate subgroup condition]] || Yes || || || If <math>H \le K \le G</math> are groups such that <math>H</math> is transfer-closed characteristic in <math>G</math>, then <math>H</math> is transfer-closed characteristic in <math>K</math>.
-{{transfercondn}}
+|-
+| [[satisfies metaproperty::strongly intersection-closed subgroup property]] || Yes || || || If <math>H_i, i \in I</math> is a family of transfer-closed characteristic subgroups of a group <math>G</math>, the intersection <math>\bigcap_{i \in I} H_i</math> is also a transfer-closed characteristic subgroup of <math>G</math>.
+|-
+| [[satisfies metaproperty::transfer condition]] || Yes || || || If <math>H</math> and <math>K</math> are subgroups of <math>G</math> such that <math>H</math> is transfer-closed characteristic in <math>G</math>, then <math>H \cap K</math> is transfer-closed characteristic in <math>K</math>.
+|}