Finitely generated solvable group: Difference between revisions
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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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| [[Stronger than:: | | [[Stronger than::finitely generated group]] || || || || {{intermediate notions short|finitely generated group|finitely generated solvable group}} | ||
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| [[Stronger than:: | | [[Stronger than::solvable group]] || || || || {{intermediate notions short|solvable group|finitely generated solvable group}} | ||
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| [[Stronger than::Hopfian group]] || any [[surjective endomorphism]] of the group is an automorphism || || || {{intermediate notions short|Hopfian group|finitely generated solvable group}} | |||
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Revision as of 01:15, 3 April 2013
This page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: finitely generated group and solvable group
View other group property conjunctions OR view all group properties
Definition
A finitely generated solvable group is a group that is both a finitely generated group and a solvable group.
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| finitely generated abelian group | |FULL LIST, MORE INFO | |||
| finitely generated nilpotent group | |FULL LIST, MORE INFO | |||
| finite solvable group | |FULL LIST, MORE INFO | |||
| polycyclic group | solvable and every subgroup is finitely generated | finitely generated and solvable not implies polycyclic | |FULL LIST, MORE INFO | |
| finitely presented solvable group | solvable and has a finite presentation | finitely generated and solvable not implies finitely presented | |FULL LIST, MORE INFO |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| finitely generated group | |FULL LIST, MORE INFO | |||
| solvable group | |FULL LIST, MORE INFO | |||
| Hopfian group | any surjective endomorphism of the group is an automorphism | |FULL LIST, MORE INFO |