# Difference between revisions of "Normal subgroup having no nontrivial homomorphism to its quotient group"

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A [[subgroup]] <math>N</math> of a [[group]] <math>G</math> is a '''normal subgroup having no nontrivial homomorphism to its quotient group''' if <math>N</math> is a [[normal subgroup]] of <math>G</math> and there is no nontrivial homomorphism from <math>N</math> to its [[quotient group]] <math>G/N</math>. | A [[subgroup]] <math>N</math> of a [[group]] <math>G</math> is a '''normal subgroup having no nontrivial homomorphism to its quotient group''' if <math>N</math> is a [[normal subgroup]] of <math>G</math> and there is no nontrivial homomorphism from <math>N</math> to its [[quotient group]] <math>G/N</math>. | ||

+ | ==Metaproperties== | ||

+ | |||

+ | {| class="sortable" border="1" | ||

+ | ! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols | ||

+ | |- | ||

+ | | [[dissatisfies metaproperty::transitive subgroup property]] || No || [[no nontrivial homomorphism to quotient group is not transitive]] || It is possible to have groups <math>H \le K \le G</math> such that <math>H</math> is normal in <math>K</math> with no nontrivial homomorphism from <math>H</math> to <math>K/H</math>, and <math>K</math> is normal in <math>G</math> with no nontrivial homomorphism from <math>K</math> to <math>G/K</math>, but there does exist a nontrivial homomorphism from <math>H</math> to <math>G/H</math>. | ||

+ | |- | ||

+ | | [[satisfies metaproperty::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. | ||

+ | |- | ||

+ | | [[satisfies metaproperty::intermediate subgroup condition]] || Yes || [[no nontrivial homomorphism to quotient group satisfies intermediate subgroup condition]] || If <math>H \le K \le G</math> are groups such that <math>H</math> is normal in <math>G</math> and there is no nontrivial homomorphism from <math>H</math> to <math>G/H</math>, <math>H</math> is also normal in <math>K</math> with no nontrivial homomorphism from <math>H</math> to <math>K/H</math>. | ||

+ | |} | ||

==Relation with other properties== | ==Relation with other properties== | ||

===Stronger properties=== | ===Stronger properties=== | ||

− | + | {| class="sortable" border="1" | |

− | + | ! Property !! Meaning !! Proof of implication !! Proof of strictness !! Intermediate notions | |

− | + | |- | |

+ | | [[Weaker than::normal Sylow subgroup]] || the whole group is a [[finite group]] and the subgroup is both normal and a [[Sylow subgroup]].|| || || {{intermediate notions short|normal subgroup having no nontrivial homomorphism to its quotient group|normal Sylow subgroup}} | ||

+ | |- | ||

+ | | [[Weaker than::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. || || || {{intermediate notions short|normal subgroup having no nontrivial homomorphism to its quotient group|normal Hall subgroup}} | ||

+ | |- | ||

+ | | [[Weaker than::fully invariant direct factor]] || both a [[fully invariant subgroup]] and a [[direct factor]] of the whole group. || || || {{intermediate notions short|normal subgroup having no nontrivial homomorphism to its quotient group|fully invariant direct factor}} | ||

+ | |} | ||

===Weaker properties=== | ===Weaker properties=== | ||

− | + | {| class="sortable" border="1" | |

− | + | ! Property !! Meaning !! Proof of implication !! Proof of strictness !! Intermediate notions | |

− | + | |- | |

− | + | | [[Stronger than::homomorph-containing subgroup]] || contains every homomorphic image of itself in the whole group. || || [[homomorph-containing not implies no nontrivial homomorphism to quotient group]] || {{intermediate notions short|homomorph-containing subgroup|normal subgroup having no nontrivial homomorphism to its quotient group}} | |

+ | |- | ||

+ | | [[Stronger than::quotient-subisomorph-containing subgroup]] || || || || {{intermediate notions short|quotient-subisomorph-containing subgroup|normal subgroup having no nontrivial homomorphism to its quotient group}} | ||

+ | |- | ||

+ | | [[Stronger than::fully invariant subgroup]] || invariant under all [[endomorphism]]s || (via homomorph-containing) || (via homomorph-containing) || {{intermediate notions short|fully invariant subgroup|normal subgroup having no nontrivial homomorphism to its quotient group}} | ||

+ | |- | ||

+ | | [[Stronger than::characteristic subgroup]] || invariant under all [[automorphism]]s || (via fully invariant) || (via fully invariant) || {{intermediate notions short|characteristic subgroup|normal subgroup having no nontrivial homomorphism to its quotient group}} | ||

+ | |- | ||

+ | | [[Stronger than::normal subgroup]] || || (by definition) || (via characteristic) || {{intermediate notions short|normal subgroup|normal subgroup having no nontrivial homomorphism to its quotient group}} | ||

+ | |} | ||

===Related properties=== | ===Related properties=== |

## Latest revision as of 20:11, 16 February 2013

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

### 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`