Closure-characteristic subgroup

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]

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Definition

Symbol-free definition

A subgroup of a group is termed closure-characteristic if its normal closure in the whole group is a characteristic subgroup.

Definition with symbols

A subgroup $H$ of a group $G$ is termed closure-characteristic if the normal closure $H^G$ of $H$ in $G$ is characteristic in $G$.

Relation with other properties

Stronger properties

• Characteristic subgroup
• Automorph-conjugate subgroup
• Join of automorph-conjugate subgroups
• Sylow subgroup
• Hall subgroup

Conjunction with other properties

Any normal subgroup that is also closure-characteristic, is characteristic.

Metaproperties

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

Join-closedness

YES: This subgroup property is join-closed: an arbitrary (nonempty) join of subgroups with this property, also has this property.
ABOUT THIS PROPERTY: View variations of this property that are join-closed | View variations of this property that are not join-closed
ABOUT JOIN-CLOSEDNESS: View all join-closed subgroup properties (or, strongly join-closed properties) | View all subgroup properties that are not join-closed | Read a survey article on proving join-closedness | Read a survey article on disproving join-closedness