Group with nilpotent derived subgroup: Difference between revisions
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==Definition== | ==Definition== | ||
A '''group with nilpotent commutator subgroup''' is a group | A '''group with nilpotent commutator subgroup''', also called a '''nilpotent-by-abelian group''', is a group satisfying the following equivalent conditions: | ||
* Its [[fact about::commutator subgroup]] is a [[fact about::nilpotent group]]. | |||
* It has a [[fact about::nilpotent normal subgroup]] with an abelian quotient group. | |||
* It has a [[fact about::nilpotent characteristic subgroup]] with an abelian quotient group. | |||
==Relation with other properties== | ==Relation with other properties== | ||
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* [[Weaker than::Supersolvable group]] | * [[Weaker than::Supersolvable group]] | ||
* [[Weaker than::Nilpotent group]] | * [[Weaker than::Nilpotent group]] | ||
* [[Weaker than::Metabelian group]] | |||
===Weaker properties=== | ===Weaker properties=== | ||
* [[Stronger than::Group satisfying subnormal join property]]: {{proofofstrictimplicationat|[[Nilpotent commutator subgroup implies subnormal join property]]|[[Subnormal join property not implies nilpotent commutator subgroup]]}} | * [[Stronger than::Group satisfying subnormal join property]]: {{proofofstrictimplicationat|[[Nilpotent commutator subgroup implies subnormal join property]]|[[Subnormal join property not implies nilpotent commutator subgroup]]}} | ||
Revision as of 18:25, 14 February 2009
This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
View a complete list of group properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions
Definition
A group with nilpotent commutator subgroup, also called a nilpotent-by-abelian group, is a group satisfying the following equivalent conditions:
- Its Commutator subgroup (?) is a Nilpotent group (?).
- It has a Nilpotent normal subgroup (?) with an abelian quotient group.
- It has a Nilpotent characteristic subgroup (?) with an abelian quotient group.
Relation with other properties
Stronger properties
Weaker properties
- Group satisfying subnormal join property: For proof of the implication, refer Nilpotent commutator subgroup implies subnormal join property and for proof of its strictness (i.e. the reverse implication being false) refer Subnormal join property not implies nilpotent commutator subgroup.