Finitely generated solvable group: Difference between revisions
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| [[Weaker than::finite solvable group]] || || || || {{intermediate notions short|finitely generated solvable group|finite solvable group}} | | [[Weaker than::finite solvable group]] || || || || {{intermediate notions short|finitely generated solvable group|finite solvable group}} | ||
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| [[Weaker than::polycyclic group]] || solvable and every subgroup is finitely generated || || || {{intermediate notions short|finitely generated solvable group|polycyclic group}} | | [[Weaker than::polycyclic group]] || solvable and every subgroup is finitely generated || || [[finitely generated and solvable not implies polycyclic]] || {{intermediate notions short|finitely generated solvable group|polycyclic group}} | ||
|- | |||
| [[Weaker than::finitely presented solvable group]] || solvable and has a finite presentation || || [[finitely generated and solvable not implies finitely presented]] || {{intermediate notions short|finitely generated solvable group|finitely presented solvable group}} | |||
|} | |} | ||
Revision as of 01:28, 29 March 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 |