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.
 
==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]]

Revision as of 20:12, 18 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.

Relation with other properties

Stronger properties

Weaker properties

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