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

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(Weaker properties)
 
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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>.
  
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==Metaproperties==
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{| class="sortable" border="1"
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! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols
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|-
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| [[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>.
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|-
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| [[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.
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|-
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| [[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>.
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|}
 
==Relation with other properties==
 
==Relation with other properties==
  
 
===Stronger properties===
 
===Stronger properties===
  
* [[Weaker than::Normal Sylow subgroup]]
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{| class="sortable" border="1"
* [[Weaker than::Normal Hall subgroup]]
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! Property !! Meaning !! Proof of implication !! Proof of strictness !! Intermediate notions
* [[Weaker than::Fully invariant direct factor]]
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|-
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| [[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}}
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|-
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| [[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}}
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|-
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| [[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}}
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|}
  
 
===Weaker properties===
 
===Weaker properties===
  
* [[Stronger than::Homomorph-containing subgroup]]
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{| class="sortable" border="1"
* [[Stronger than::Quotient-homomorph-containing subgroup]]
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! Property !! Meaning !! Proof of implication !! Proof of strictness !! Intermediate notions
* [[Stronger than::Fully invariant subgroup]]
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|-
* [[Stronger than::Characteristic subgroup]]
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| [[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}}
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|-
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| [[Stronger than::quotient-subisomorph-containing subgroup]] || || || || {{intermediate notions short|quotient-subisomorph-containing subgroup|normal subgroup having no nontrivial homomorphism to its quotient group}}
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|-
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| [[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}}
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|-
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| [[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}}
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|-
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| [[Stronger than::normal subgroup]] || || (by definition) || (via characteristic) || {{intermediate notions short|normal subgroup|normal subgroup having no nontrivial homomorphism to its quotient group}}
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|}
  
 
===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]

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.

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 H \le K \le G such that H is normal in K with no nontrivial homomorphism from H to K/H, and K is normal in G with no nontrivial homomorphism from K to G/K, but there does exist a nontrivial homomorphism from H to G/H.
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 H \le K \le G are groups such that H is normal in G and there is no nontrivial homomorphism from H to G/H, H is also normal in K with no nontrivial homomorphism from H to K/H.

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) Fully invariant subgroup, 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) Fully invariant subgroup, Homomorph-containing subgroup|FULL LIST, MORE INFO

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