Characteristically complemented subgroup: Difference between revisions

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===Symbol-free definition===
===Symbol-free definition===


A [[subgroup]] of a [[group]] is termed a '''characteristic retract''' if there is a [[retraction]] (viz an idempotent endomorphism) on the group, whose image is that subgroup, and whose kernel is a [[characteristic subgroup]].
A [[subgroup]] of a [[group]] is termed a '''characteristically complemented subgroup''' or '''characteristic retract''' if it satisfies the following equivalent conditions:
 
# There is a [[defining ingredient::retraction]] (viz an idempotent endomorphism) on the group, whose image is that subgroup, and whose kernel is a [[defining ingredient::characteristic subgroup]].
# There is a [[characteristic subgroup]] that is a [[defining ingredient::permutable complements|permutable complement]] to it.
# There is a [[characteristic subgroup]] that is a [[defining ingredient::lattice complements|lattice complement]] to it.
 
The corresponding characteristic subgroup (there may be more than one such) is termed a [[complemented characteristic subgroup]].
 
==Relation with other properties==
 
===Stronger properties===
 
* [[Weaker than::Characteristically complemented characteristic subgroup]]
* [[Weaker than::Characteristically complemented normal subgroup]]
 
===Weaker properties===
 
* [[Stronger than::Quasicharacteristic retract]]
* [[Stronger than::Retract]]
* [[Stronger than::Permutably complemented subgroup]]
* [[Stronger than::Lattice-complemented subgroup]]
 
==Metaproperties==
 
{{transitive}}
 
If <math>H \le K \le G</math> are groups such that <math>H</math> is characteristically complemented in <math>K</math> and <math>K</math> is characteristically complemented in <math>G</math>, then <math>H</math> is characteristically complemented in <math>G</math>. This follows essentially from the fact that [[characteristicity is quotient-transitive]]. {{proofat|[[Characteristically complemented is transitive]]}}
 
{{further|[[Characteristicity is quotient-transitive]], [[Quotient-transitive and stronger than normality implies complementary property is transitive]]}}
{{quot-transitive}}

Latest revision as of 18:52, 20 May 2009

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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]

Definition

Symbol-free definition

A subgroup of a group is termed a characteristically complemented subgroup or characteristic retract if it satisfies the following equivalent conditions:

  1. There is a retraction (viz an idempotent endomorphism) on the group, whose image is that subgroup, and whose kernel is a characteristic subgroup.
  2. There is a characteristic subgroup that is a permutable complement to it.
  3. There is a characteristic subgroup that is a lattice complement to it.

The corresponding characteristic subgroup (there may be more than one such) is termed a complemented characteristic subgroup.

Relation with other properties

Stronger properties

Weaker properties

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

If H≤K≤G are groups such that H is characteristically complemented in K and K is characteristically complemented in G, then H is characteristically complemented in G. This follows essentially from the fact that characteristicity is quotient-transitive. For full proof, refer: Characteristically complemented is transitive

Further information: Characteristicity is quotient-transitive, Quotient-transitive and stronger than normality implies complementary property is transitive

Quotient-transitivity

This subgroup property is quotient-transitive: the corresponding quotient property is transitive.
View a complete list of quotient-transitive subgroup properties

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