Fully invariant closure: Difference between revisions

Latest revision as of 00:45, 12 January 2010

This article defines a subgroup operator related to the subgroup property fully invariant subgroup. By subgroup operator is meant an operator that takes as input a subgroup of a group and outputs a subgroup of the same group.

This article is about a definition in group theory that is standard among the group theory community (or sub-community that dabbles in such things) but is not very basic or common for people outside.
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Definition

Symbol-free definition

The fully invariant closure of a subgroup in a group can be defined in any of the following equivalent ways:

As the intersection of all fully invariant subgroups containing the given subgroup
As the subgroup generated by all endomorphs to the given subgroup
As the set of all elements that can be written as products of finite length of elements from the subgroup and their endomorphic images

Definition with symbols

The fully invariant closure of a subgroup $H$ in a group $G$ , is defined in the following equivalent ways:

As the intersection of all fully invariant subgroups of $G$ containing $H$
As the subgroup generated by all $\rho (H)$ where $\rho \in \operatorname {End} (G)$

Relation with other operators

Related operators

@@ Line 1: / Line 1: @@
-{{subgroupoperatorrelatedto|full characteristicity}}
+{{subgroupoperatorrelatedto|fully invariant subgroup}}
 {{stdnonbasicdef}}
@@ Line 7: / Line 7: @@
 ===Symbol-free definition===
-The '''fully characteristic closure''' of a subgroup  in a [[group]] can be defined in any of the following equivalent ways:
+The '''fully invariant closure''' of a subgroup  in a [[group]] can be defined in any of the following equivalent ways:
-* As the intersection of all [[fully characteristic subgroup]]s containing the given subgroup
+* As the intersection of all [[defining ingredient::fully invariant subgroup]]s containing the given subgroup
-* As the subgroup generated by all [[endomorph]]s to the given subgroup
+* As the subgroup generated by all [[defining ingredient::endomorph]]s to the given subgroup
 * As the set of all elements that can be written as products of finite length of elements from the subgroup and their endomorphic images
 ===Definition with symbols===
-The '''fully characteristic closure''' of a [[subgroup]] <math>H</math> in a [[group]] <math>G</math>, is defined in the following equivalent ways:
+The '''fully invariant closure''' of a [[subgroup]] <math>H</math> in a [[group]] <math>G</math>, is defined in the following equivalent ways:
-* As the intersection of all [[fully characteristic subgroup]]s of <math>G</math> containing <math>H</math>
+* As the intersection of all [[fully invariant subgroup]]s of <math>G</math> containing <math>H</math>
-* As the subgroup generated by all <math>\rho(H)</math> where <math>\rho \in End(G)</math>
+* As the subgroup generated by all <math>\rho(H)</math> where <math>\rho \in \operatorname{End}(G)</math>
 ==Relation with other operators==