# 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
fully invariant subgroup invariant under all endomorphisms |FULL LIST, MORE INFO
endomorph-dominating subgroup image under any endomorphism is contained in a conjugate subgroup to it |FULL LIST, MORE INFO
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
join of Sylow subgroups join of Sylow subgroups Subgroup whose normal closure is homomorph-containing|FULL LIST, MORE INFO
Hall subgroup subgroup whose order and index are relatively prime (via join of Sylow subgroups) (via join of Sylow subgroups) Join of Sylow subgroups, Subgroup whose normal closure is homomorph-containing|FULL LIST, MORE INFO
subgroup whose normal closure is homomorph-containing normal closure is a homomorph-containing subgroup follows from homomorph-containing implies fully invariant follows from fully invariant not implies homomorph-containing, and the fact that, since fully invariant implies normal, a fully invariant subgroup equals its own normal closure |FULL LIST, MORE INFO
contranormal subgroup normal closure is the whole group |FULL LIST, MORE INFO

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
closure-characteristic subgroup normal closure is a characteristic subgroup fully invariant implies characteristic follows by combining characteristic not implies fully invariant and the fact that, since characteristic implies normal, a characteristic subgroup is its own normal closure |FULL LIST, MORE INFO
subgroup whose characteristic closure is fully invariant characteristic closure is a fully invariant subgroup if the normal closure is fully invariant, it is also characteristic (because fully invariant implies characteristic), and since characteristic implies normal, it is the characteristic closure proper nontrivial subgroup in a characteristically simple group, e.g., Z2 in V4 |FULL LIST, MORE INFO