CEP-subgroup: Difference between revisions
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===Symbol-free definition=== | ===Symbol-free definition=== | ||
A subgroup of a group is termed a '''CEP subgroup''' (or a group with '''Congruence Extension Property''') if any normal subgroup of this subgroup is the intersection of this subgroup with a normal subgroup of the whole group. | A [[subgroup]] of a [[group]] is termed a '''CEP subgroup''' (or a group with '''Congruence Extension Property''') if any [[normal subgroup]] of this subgroup is the intersection of this subgroup with a [[normal subgroup]] of the whole group. | ||
CEP-subgroups are also termed '''normal-sensitive subgroups'''. | CEP-subgroups are also termed '''normal-sensitive subgroups'''. | ||
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===Definition with symbols=== | ===Definition with symbols=== | ||
A subgroup <math>H</math> of a group <math>G</math> is termed a '''CEP subgroup''' if for any normal subgroup <math>N</math> of <math>H</math>, there is a normal subgroup <math>M</math> of <math>G</math> such that <math>N = M</math> ∩ <math>H</math>. | A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed a '''CEP subgroup''' if for any [[normal subgroup]] <math>N</math> of <math>H</math>, there is a normal subgroup <math>M</math> of <math>G</math> such that <math>N = M</math> ∩ <math>H</math>. | ||
==Formalisms== | ==Formalisms== | ||
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===Stronger properties=== | ===Stronger properties=== | ||
* [[Weaker than::Retract]] | |||
* [[Weaker than::Transitively normal subgroup]] | |||
===Weaker properties=== | |||
* [[Stronger than::Subgroup in which every relatively normal subgroup is strongly closed]] | |||
* [[Stronger than::Subgroup in which every relatively normal subgroup is weakly closed]] | |||
==Metaproperties== | ==Metaproperties== | ||
Latest revision as of 21:54, 11 February 2009
This article defines a subgroup property related to (or which arises in the context of): geometric group theory
View other subgroup properties related to geometric group theory|View other terms related to geometric group theory | View all subgroup properties
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 CEP-subgroup, all facts related to CEP-subgroup) |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
A subgroup of a group is termed a CEP subgroup (or a group with Congruence Extension Property) if any normal subgroup of this subgroup is the intersection of this subgroup with a normal subgroup of the whole group.
CEP-subgroups are also termed normal-sensitive subgroups.
Definition with symbols
A subgroup of a group is termed a CEP subgroup if for any normal subgroup of , there is a normal subgroup of such that ∩ .
Formalisms
In terms of the subgroup intersection extension formalism
In terms of the subgroup intersection extension formalism, the property of being a CEP subgroup is the balanced property with respect to the subgroup property of normality.
Facts
Every subgroup of a group is a CEP-subgroup if and only if the group is a solvable T-group.
Relation with other properties
Stronger properties
Weaker properties
- Subgroup in which every relatively normal subgroup is strongly closed
- Subgroup in which every relatively normal subgroup is weakly closed
Metaproperties
Transitivity
This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property in the whole group.
ABOUT THIS PROPERTY: View variations of this property that are transitive | View variations of this property that are not transitive
ABOUT TRANSITIVITY: View a complete list of transitive subgroup properties|View a complete list of facts related to transitivity of subgroup properties |Read a survey article on proving transitivity
Any CEP subgroup of a CEP subgroup is a CEP subgroup. This follows from the fact that the property of being CEP is a balanced subgroup property with respect to a suitable formalism.
Trimness
This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself).
View other trim subgroup properties | View other trivially true subgroup properties | View other identity-true subgroup properties
The improper subgroup, or the whole group, is clearly a CEP subgroup, so the property of being a CEP subgroup is identity-true.
The trivial subgroup is also a CEP subgroup, so the property of being a CEP subgroup is trivially true.
Thus, the property of being a CEP subgroup is a trim subgroup property.