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

From Groupprops

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* [[Complemented normal subgroup]]: {{proofat|[[No nontrivial homomorphism to quotient group not implies complemented normal]]}} | * [[Complemented normal subgroup]]: {{proofat|[[No nontrivial homomorphism to quotient group not implies complemented normal]]}} | ||

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+ | ==Metaproperties== | ||

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+ | {{intransitive}} | ||

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+ | {{proofat|[[No nontrivial homomorphism to quotient group is not transitive]]}} | ||

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+ | {{trim}} | ||

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+ | {{intsubcondn}} | ||

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+ | If <math>H \le K \le G</math>, with <math>H</math> normal in <math>G</math> and 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>. | ||

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+ | {{proofat|[[No nontrivial homomorphism to quotient group satisfies intermediate subgroup condition]]}} |

## Revision as of 21:36, 13 August 2009

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 .

## Relation with other properties

### Stronger properties

### Weaker properties

- Homomorph-containing subgroup
- Quotient-homomorph-containing subgroup
- Fully invariant subgroup
- Characteristic subgroup

### 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`