Finitely generated Hopfian group: Difference between revisions
(Created page with '{{group property conjunction|finitely generated group|Hopfian group}} ==Definition== A '''finitely generated Hopfian group''' is a group that is both a [[finitely generated...') |
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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 | ||
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| [[Weaker than::Finite group]] || || || || {{intermediate notions short|finitely generated Hopfian group|finite group}} | |||
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| [[Weaker than::Finitely generated residually finite group]] || finitely generated and [[residually finite group|residually finite]] || [[finitely generated and residually finite implies Hopfian]] || || {{intermediate notions short|finitely generated Hopfian group|finitely generated residually finite group}} | | [[Weaker than::Finitely generated residually finite group]] || finitely generated and [[residually finite group|residually finite]] || [[finitely generated and residually finite implies Hopfian]] || || {{intermediate notions short|finitely generated Hopfian group|finitely generated residually finite group}} | ||
Latest revision as of 16:14, 25 May 2010
This page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: finitely generated group and Hopfian group
View other group property conjunctions OR view all group properties
Definition
A finitely generated Hopfian group is a group that is both a finitely generated group (i.e., it has a finite generating set) and a Hopfian group (i.e., every surjective endomorphism of the group is an automorphism).
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| Finite group | |FULL LIST, MORE INFO | |||
| Finitely generated residually finite group | finitely generated and residually finite | finitely generated and residually finite implies Hopfian | |FULL LIST, MORE INFO | |
| Finitely generated free group | finitely generated and a free group; equivalently, free on a finite freely generating set | finitely generated and free implies Hopfian | |FULL LIST, MORE INFO | |
| Finitely generated abelian group | finitely generated and an abelian group | finitely generated abelian implies Hopfian | |FULL LIST, MORE INFO | |
| Slender group | every subgroup is a finitely generated group | slender implies Hopfian, (slender implies finitely generated by definition) | |FULL LIST, MORE INFO | |
| Finitely generated simple group | finitely generated and a simple group | simple implies Hopfian | |FULL LIST, MORE INFO |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| Hopfian group | |FULL LIST, MORE INFO | |||
| Finitely generated group | |FULL LIST, MORE INFO |