Finitely generated solvable group: Difference between revisions
(Created page with '{{group property conjunction|finitely generated group|solvable group}} ==Definition== A '''finitely generated solvable group''' is a group that is both a [[finitely generat...') |
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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:: | | [[Weaker than::finitely generated abelian group]] || || || || {{intermediate notions short|finitely generated solvable group|finitely generated abelian group}} | ||
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| [[Weaker than:: | | [[Weaker than::finitely generated nilpotent group]] || || || || {{intermediate notions short|finitely generated solvable group|finitely generated nilpotent group}} | ||
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| [[Weaker than:: | | [[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}} | |||
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Revision as of 21:51, 22 January 2012
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 | |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 |