# Subgroup whose normal closure is homomorph-containing

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]

## Contents

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

## Definition

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

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
homomorph-dominating subgroup any subgroup of the whole group that is a homomorphic image of the subgroup is contained in one of its conjugates Join of homomorph-dominating subgroups|FULL LIST, MORE INFO
Sylow subgroup subgroup of finite group with maximal prime power order (via homomorph-dominating) Join of Sylow subgroups|FULL LIST, MORE INFO
join of Sylow subgroups join of Sylow subgroups follows from statement for Sylow subgroups, and homomorph-containment is strongly join-closed Join of homomorph-dominating subgroups|FULL LIST, MORE INFO
Hall subgroup order and index are relatively prime (via join of Sylow subgroups) Join of Sylow subgroups, Join of homomorph-dominating subgroups|FULL LIST, MORE INFO

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
subgroup whose normal closure is fully invariant normal closure is a fully invariant 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 is its own normal closure |FULL LIST, MORE INFO
subgroup whose normal closure is characteristic normal closure is a characteristic subgroup (via subgroup whose normal closure is fully invariant, using fully invariant implies characteristic) (via subgroup whose normal closure is fully invariant) Subgroup whose normal closure is fully invariant|FULL LIST, MORE INFO