Normality-large normal subgroup

From Groupprops
Jump to: navigation, search
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:

  1. Its intersection with every nontrivial normal subgroup is a nontrivial normal subgroup
  2. 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.