Normality-large normal subgroup: Difference between revisions
(New page: {{subgroup property conjunction|normality-large subgroup|normal subgroup}} ==Definition== A '''normality-large normal subgroup''' is a subgroup that satisfies the following equivalent co...) |
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# Its intersection with every [[nontrivial normal subgroup]] is a nontrivial normal subgroup | # Its intersection with every [[nontrivial normal subgroup]] is a nontrivial normal subgroup | ||
# It is [[normal subgroup|normal]] and [[normality-large subgroup|normality-large]]: its intersection with every nontrivial normal subgroup is nontrivial | # It is [[normal subgroup|normal]] and [[normality-large subgroup|normality-large]]: its intersection with every nontrivial normal subgroup is nontrivial | ||
For a [[finite group]] (or more generally a [[group in which every nontrivial normal subgroup contains a minimal normal subgroup]]), there is | For a [[finite group]] (or more generally a [[group in which every nontrivial normal subgroup contains a minimal normal subgroup]]), there is another equivalent condition: it is a [[normal subgroup]] containing the [[socle]]. | ||
==Metaproperties== | ==Metaproperties== | ||
Latest revision as of 18:16, 13 July 2008
This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: normality-large subgroup and normal subgroup
View other subgroup property conjunctions | view all subgroup properties
Definition
A normality-large normal subgroup is a subgroup that satisfies the following equivalent conditions:
- Its intersection with every nontrivial normal subgroup is a nontrivial normal subgroup
- It is normal and normality-large: its intersection with every nontrivial normal subgroup is nontrivial
For a finite group (or more generally a group in which every nontrivial normal subgroup contains a minimal normal subgroup), there is another equivalent condition: it is a normal subgroup containing the socle.
Metaproperties
Intersection-closedness
This subgroup property is finite-intersection-closed; a finite (nonempty) intersection of subgroups with this property, also has this property
View a complete list of finite-intersection-closed subgroup properties
A finite intersection of normality-large normal subgroups is normality-large normal.