Normal closure-closed subgroup property

From Groupprops
Jump to: navigation, search
This article defines a subgroup metaproperty: a property that can be evaluated to true/false for any subgroup property
View a complete list of subgroup metaproperties
View subgroup properties satisfying this metaproperty| View subgroup properties dissatisfying this metaproperty
VIEW RELATED: subgroup metaproperty satisfactions| subgroup metaproperty dissatisfactions

Definition

Symbol-free definition

A subgroup property is said to be normal closure-closed if whenever a subgroup has the property in the whole group, its normal closure also has the property.

Definition with symbols

A subgroup property p is said to be normal core-closed if whenever H satisfies property p in G, the normal core <H^G> also satisfies p in G.

Relation with other metaproperties

Stronger metaproperties

Related metaproperties

Opposite metaproperties

Subgroup properties that are normal closure-closed

Join-closed subgroup properties

One way of establishing that a subgroup property is normal closure-closed is to show that the join of any family of subgroups having the property, also has the property.

Subgroup properties stronger than normality

If a subgroup property is stronger than normality, then the normal closure of any subgroup having the property is itself, and hence whenever a subgroup has the property, so does its normal closure.