Hopfian group: Difference between revisions
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[[Importance rank::3| ]] | |||
{{group property}} | {{group property}} | ||
{{variation of|finite group}} | {{variation of|finite group}} | ||
{{finitarily tautological group property}} | {{finitarily tautological group property}} | ||
==Definition== | ==Definition== | ||
A [[group]] is termed '''Hopfian''' if it satisfies the following equivalent conditions: | A [[group]] is termed '''Hopfian''' if it satisfies the following equivalent conditions: | ||
* It is not isomorphic to the [[quotient group]] by any nontrivial [[normal subgroup]] (in short, it is not isomorphic to any of its proper quotients). | * It is not isomorphic to the [[quotient group]] by any nontrivial [[normal subgroup]] (in short, it is not isomorphic to any of its proper quotients). | ||
* Every [[surjective endomorphism]] of it is an [[automorphism]] | * Every [[surjective endomorphism]] of it is an [[automorphism]]. | ||
=== | ==Metaproperties== | ||
{| class="sortable" border="1" | |||
! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols | |||
|- | |||
| [[dissatisfies metaproperty::subgroup-closed group property]] || No || [[Hopfianness is not subgroup-closed]] || It is possible to have a Hopfian group <math>G</math> and a subgroup <math>H \le G</math> such that <math>H</matH> is not Hopfian. | |||
|- | |||
| [[dissatisfies metaproperty::quotient-closed group property]] || No || [[Hopfianness is not quotient-closed]] || It is possible to have a Hopfian group <math>G</math> and a normal subgroup <math>H</math> of <math>G</math> such that the [[quotient group]] <math>G/H</math> is not Hopfian. | |||
|} | |||
==Relation with other properties== | ==Relation with other properties== | ||
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! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | ! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | ||
|- | |- | ||
| [[Weaker than:: | | [[Weaker than::finite group]] || [[order of a group|order]] is finite || || || {{intermediate notions short|Hopfian group|finite group}} | ||
|- | |- | ||
| [[Weaker than:: | | [[Weaker than::finitely generated free group]] || [[free group]] on finite [[generating set]] || [[finitely generated and free implies Hopfian]] || || {{intermediate notions short|Hopfian group|finitely generated free group}} | ||
|- | |- | ||
| [[Weaker than:: | | [[Weaker than::finitely generated residually finite group]] || [[finitely generated group|finitely generated]] and [[residually finite group|residually finite]] || [[finitely generated and residually finite implies Hopfian]] || || {{intermediate notions short|Hopfian group|finitely generated residually finite group}} | ||
|- | |- | ||
| [[Weaker than:: | | [[Weaker than::group satisfying ascending chain condition on normal subgroups]] || there is no infinite strictly ascending chain of [[normal subgroup]]s || [[ascending chain condition on normal subgroups implies Hopfian]] || || {{intermediate notions short|Hopfian group|group satisfying ascending chain condition on normal subgroups}} | ||
|- | |- | ||
| [[Weaker than::Noetherian group]] (also called '''slender group''') || every subgroup is finitely generated; equivalently, no infinite strictly ascending chain of subgroups || [[ | | [[Weaker than::Noetherian group]] (also called '''slender group''') || every subgroup is finitely generated; equivalently, no infinite strictly ascending chain of subgroups || [[Noetherian implies Hopfian]] (proof is [[ascending chain condition on normal subgroups implies Hopfian|ascending chain condition on normal subgroups]]) || || {{intermediate notions short|Hopfian group|slender group}} | ||
|- | |- | ||
| [[Weaker than:: | | [[Weaker than::finitely generated Hopfian group]] || [[finitely generated group|finitely generated]] and Hopfian || (by definition) || [[Hopfian not implies finitely generated]] || {{intermediate notions short|Hopfian group|finitely generated Hopfian group}} | ||
|- | |- | ||
| [[Weaker than:: | | [[Weaker than::simple group]] || has no proper nontrivial [[normal subgroup]]s || [[simple implies Hopfian]] || || {{intermediate notions short|Hopfian group|simple group}} | ||
|- | |- | ||
| [[Weaker than:: | | [[Weaker than::group in which every endomorphism is trivial or an automorphism]] || every [[endomorphism]] is either trivial or is an [[automorphism]] || [[every endomorphism is trivial or an automorphism implies Hopfian]] || || {{intermediate notions short|Hopfian group|group in which every endomorphism is trivial or an automorphism}} | ||
|- | |- | ||
| [[Weaker than::Group in which every endomorphism is trivial or injective]] || every [[endomorphism]] is either trivial or an [[injective endomorphism]] || [[every endomorphism is trivial or injective implies Hopfian]] || (finite counterexamples) || {{intermediate notions short|Hopfian group|group in which every endomorphism is trivial or injective}} | | [[Weaker than::Group in which every endomorphism is trivial or injective]] || every [[endomorphism]] is either trivial or an [[injective endomorphism]] || [[every endomorphism is trivial or injective implies Hopfian]] || (finite counterexamples) || {{intermediate notions short|Hopfian group|group in which every endomorphism is trivial or injective}} | ||
|- | |||
| [[Weaker than::finitely generated abelian group]] || || [[finitely generated abelian implies Hopfian]] (also via Noetherian) || any finite non-abelian group || {{intermediate notions short|Hopfian group|finitely generated abelian group}} | |||
|- | |||
| [[Weaker than::finitely generated nilpotent group]] || || [[finitely generated and nilpotent implies Hopfian]] ([[equivalence of definitions of finitely generated nilpotent group|via Noetherian]]) || (via Noetherian, also any finite non-nilpotent counterexample) || {{intermediate notions short|Hopfian group|finitely generated nilpotent group}} | |||
|} | |} | ||
Latest revision as of 04:33, 3 April 2013
This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
View a complete list of group properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions
This is a variation of finite group|Find other variations of finite group |
This property makes sense for infinite groups. For finite groups, it is always true
Definition
A group is termed Hopfian if it satisfies the following equivalent conditions:
- It is not isomorphic to the quotient group by any nontrivial normal subgroup (in short, it is not isomorphic to any of its proper quotients).
- Every surjective endomorphism of it is an automorphism.
Metaproperties
| Metaproperty name | Satisfied? | Proof | Statement with symbols |
|---|---|---|---|
| subgroup-closed group property | No | Hopfianness is not subgroup-closed | It is possible to have a Hopfian group and a subgroup such that is not Hopfian. |
| quotient-closed group property | No | Hopfianness is not quotient-closed | It is possible to have a Hopfian group and a normal subgroup of such that the quotient group is not Hopfian. |
