Right-transitively permutable subgroup: Difference between revisions
(New page: {{wikilocal}} {{subgroup property}} ==Definition== ===Symbol-free definition=== A subgroup of a group is termed '''right-transitively permutable''' if any [[permutable subgroup]...) |
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A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed '''right-transitively permutable''' if whenever <math>K</math> is a [[permutable subgroup]] of <math>H</math>, then <math>K</math> is permutable in <math>G</math>. | A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed '''right-transitively permutable''' if whenever <math>K</math> is a [[permutable subgroup]] of <math>H</math>, then <math>K</math> is permutable in <math>G</math>. | ||
==Formalisms== | |||
{{obtainedbyapplyingthe|right transiter|permutable subgroup}} | |||
==Relation with other properties== | ==Relation with other properties== | ||
Latest revision as of 00:08, 8 May 2008
BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
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 right-transitively permutable if any permutable subgroup of the subgroup is also permutable in the whole group.
Definition with symbols
A subgroup of a group is termed right-transitively permutable if whenever is a permutable subgroup of , then is permutable in .
Formalisms
In terms of the right transiter
This property is obtained by applying the right transiter to the property: permutable subgroup
View other properties obtained by applying the right transiter
Relation with other properties
Stronger properties
- Hall direct factor: A Hall subgroup that is also a direct factor
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
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