# Normal subgroup having no nontrivial homomorphism from its quotient group

From Groupprops

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

## Definition

A subgroup of a group is termed a **normal subgroup having no nontrivial homomorphism from its quotient group** if is a normal subgroup of and there is no nontrivial homomorphism of groups from the quotient group to .

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

normal Sylow subgroup | the whole group is a finite group and the subgroup is normal as well as a Sylow subgroup | |FULL LIST, MORE INFO | ||

normal Hall subgroup | the whole group is a finite group and the subgroup is normal as well as a Hall subgroup -- its order and index are relatively prime. | |FULL LIST, MORE INFO |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

normal subgroup |