Transfer-closed characteristic subgroup: Difference between revisions

Latest revision as of 23:54, 31 May 2020

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 Transfer-closed characteristic subgroup, all facts related to Transfer-closed characteristic 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

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

This is a variation of characteristicity|Find other variations of characteristicity | Read a survey article on varying characteristicity

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 $K$ .

@@ Line 28: / Line 28: @@
 ==Metaproperties==
-{{transitive}}
+{| class="sortable" border="1"
+! Metaproperty name !! Satisfied? !! Proof !! Difficulty level !! Statement with symbols
-Suppose <math>H \le K \le G</math> are groups such that <math>K</math> is a transfer-closed characteristic subgroup of <math>G</math> and <math>H</math> is a transfer-closed characteristic subgroup of <math>K</math>. Then, <math>H</math> is a transfer-closed characteristic subgroup of <math>G</math>.
+|-
+| [[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>.
-{{proofat|[[Transfer-closed characteristicity is transitive]]}}
+|-
+| [[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>.
-{{further|[[Transfer condition operator preserves transitivity]], [[Characteristicity is transitive]]}}
+|-
-{{trim}}
+| [[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>.
+|-
-{{intsubcondn}}
+| [[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>.
+|}
-{{transfercondn}}