Fully invariant closure: Difference between revisions
m (moved Fully characteristic closure to Fully invariant closure) |
No edit summary |
||
| Line 1: | Line 1: | ||
{{subgroupoperatorrelatedto| | {{subgroupoperatorrelatedto|fully invariant subgroup}} | ||
{{stdnonbasicdef}} | {{stdnonbasicdef}} | ||
| Line 7: | Line 7: | ||
===Symbol-free definition=== | ===Symbol-free definition=== | ||
The '''fully | 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 | * 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 | * 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=== | ===Definition with symbols=== | ||
The '''fully | 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 | * 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 End(G)</math> | ||
Revision as of 00:00, 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.
VIEW: Definitions built on this | Facts about this: (facts closely related to Fully invariant closure, all facts related to Fully invariant closure) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
View a list of other standard non-basic definitions
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 in a group , is defined in the following equivalent ways:
- As the intersection of all fully invariant subgroups of containing
- As the subgroup generated by all where
