Subgroup whose normal closure is fully invariant

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

A subgroup $H$ of a group $G$ is termed a subgroup whose normal closure is fully invariant if $H^G$, its normal closure in the whole group, is a fully invariant subgroup of $G$.

Examples

Extreme examples

• The trivial subgroup satisfies this property in every group.
• Every group satisfies this property in itself.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Sylow subgroup $p$-subgroup of finite group whose index is relatively prime to $p$ (via endomorph-dominating) (via endomorph-dominating) Join of Sylow subgroups, Subgroup whose normal closure is homomorph-containing|FULL LIST, MORE INFO