Statements (48)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
gptkb:mathematical_concept
Group theory concept |
| gptkbp:contrastsWith |
gptkb:Solvable_group
Simple group |
| gptkbp:definedIn |
Lower central series terminates at the trivial subgroup
|
| gptkbp:example |
Abelian group
Quaternion group |
| gptkbp:generalizes |
gptkb:Solvable_group
|
| gptkbp:hasCharacteristicSubgroup |
Center
Lower central series Upper central series |
| gptkbp:hasInvariant |
Nilpotency class
|
| gptkbp:hasProperty |
Every proper subgroup is properly contained in its normalizer
Every subgroup and quotient of a nilpotent group is nilpotent Lower central series reaches trivial group in finitely many steps Nilpotency class n means lower central series terminates at n steps Upper central series reaches group in finitely many steps Infinite nilpotent group need not be direct product of p-groups Finite nilpotent group is direct product of its Sylow subgroups Every finite p-group is nilpotent Every maximal subgroup is normal Every nilpotent group is solvable Nilpotency class 1 is abelian group Not every solvable group is nilpotent Center is nontrivial for nontrivial nilpotent group Finite nilpotent group is product of its Sylow subgroups Every homomorphic image of a nilpotent group is nilpotent |
| gptkbp:hasSpecialCase |
gptkb:Trivial_group
Abelian group p-group |
| https://www.w3.org/2000/01/rdf-schema#label |
Nilpotent group
|
| gptkbp:introduced |
gptkb:Philip_Hall
|
| gptkbp:isClosedUnder |
Direct products
Extensions by nilpotent groups Quotients Subgroups |
| gptkbp:namedAfter |
Nilpotent matrix
|
| gptkbp:property |
Direct product of nilpotent groups is nilpotent
Every Sylow subgroup is normal Every subgroup is subnormal Finite p-group is nilpotent |
| gptkbp:subclassOf |
gptkb:group_of_people
|
| gptkbp:usedIn |
gptkb:Algebraic_topology
gptkb:algebra gptkb:Algebraic_geometry gptkb:Group_theory |
| gptkbp:bfsParent |
gptkb:Algebraic_groups
|
| gptkbp:bfsLayer |
7
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