# Normal subgroup having no nontrivial homomorphism to 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 a **normal subgroup having no nontrivial homomorphism to its quotient group** if is a normal subgroup of and there is no nontrivial homomorphism from to its quotient group .

## Metaproperties

Metaproperty name | Satisfied? | Proof | Statement with symbols |
---|---|---|---|

transitive subgroup property | No | no nontrivial homomorphism to quotient group is not transitive | It is possible to have groups such that is normal in with no nontrivial homomorphism from to , and is normal in with no nontrivial homomorphism from to , but there does exist a nontrivial homomorphism from to . |

trim subgroup property | Yes | (obvious reasons) | In any group, the trivial subgroup and the whole group satisfy the condition of being normal subgroups with no nontrivial homomorphism to their respective quotient groups. |

intermediate subgroup condition | Yes | no nontrivial homomorphism to quotient group satisfies intermediate subgroup condition | If are groups such that is normal in and there is no nontrivial homomorphism from to , is also normal in with no nontrivial homomorphism from to . |

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness | Intermediate notions |
---|---|---|---|---|

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

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

fully invariant direct factor | both a fully invariant subgroup and a direct factor of the whole group. | |FULL LIST, MORE INFO |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness | Intermediate notions |
---|---|---|---|---|

homomorph-containing subgroup | contains every homomorphic image of itself in the whole group. | homomorph-containing not implies no nontrivial homomorphism to quotient group | |FULL LIST, MORE INFO | |

quotient-subisomorph-containing subgroup | |FULL LIST, MORE INFO | |||

fully invariant subgroup | invariant under all endomorphisms | (via homomorph-containing) | (via homomorph-containing) | Homomorph-containing subgroup, Intermediately fully invariant subgroup, Quotient-subisomorph-containing subgroup|FULL LIST, MORE INFO |

characteristic subgroup | invariant under all automorphisms | (via fully invariant) | (via fully invariant) | Homomorph-containing subgroup, Intermediately fully invariant subgroup, Intermediately strictly characteristic subgroup, Quotient-subisomorph-containing subgroup|FULL LIST, MORE INFO |

normal subgroup | (by definition) | (via characteristic) | Homomorph-containing subgroup|FULL LIST, MORE INFO |

### Related properties

- Normal subgroup having no nontrivial homomorphism from its quotient group
- Normal subgroup having no common composition factor with its quotient group

### Other incomparable properties

- Complemented normal subgroup:
`For full proof, refer: No nontrivial homomorphism to quotient group not implies complemented normal`

## Metaproperties

### Transitivity

NO:This subgroup property isnottransitive: a subgroup with this property in a subgroup with this property, need not have the property in the whole groupABOUT THIS PROPERTY: View variations of this property that are transitive|View variations of this property that are not transitiveABOUT TRANSITIVITY: View a complete list of subgroup properties that are not transitive|View facts related to transitivity of subgroup properties | View a survey article on disproving transitivity

`For full proof, refer: No nontrivial homomorphism to quotient group is not transitive`

### 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

### Intermediate subgroup condition

YES:This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.ABOUT THIS PROPERTY: View variations of this property satisfying intermediate subgroup condition | View variations of this property not satisfying intermediate subgroup conditionABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition

If , with normal in and no nontrivial homomorphism from to , is also normal in with no nontrivial homomorphism from to .

`For full proof, refer: No nontrivial homomorphism to quotient group satisfies intermediate subgroup condition`