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

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(Relation with other properties)
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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]

Definition

A subgroup N of a group G is a normal subgroup having no nontrivial homomorphism to its quotient group if N is a normal subgroup of G and there is no nontrivial homomorphism from N to its quotient group G/N.

Relation with other properties

Stronger properties

Weaker properties

Related properties

Other incomparable properties

Metaproperties

Transitivity

NO: This subgroup property is not transitive: a subgroup with this property in a subgroup with this property, need not have the property in the whole group
ABOUT THIS PROPERTY: View variations of this property that are transitive|View variations of this property that are not transitive
ABOUT 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 condition
ABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition

If H \le K \le G, with H normal in G and no nontrivial homomorphism from H to G/H, H is also normal in K with no nontrivial homomorphism from H to K/H.

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