Strongly central series: Difference between revisions
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is termed '''strongly central''' if <math>[K_i,K_j] \le K_{i+j}</math> for every <math>i,j = 1,2,\ldots,c</math>. | is termed '''strongly central''' if <math>[K_i,K_j] \le K_{i+j}</math> for every <math>i,j = 1,2,\ldots,c</math>. | ||
(A similar definition works for transfinite series). | |||
==Examples== | ==Examples== | ||
The [[lower central series]] and [[upper central series]] of a [[nilpotent group]] are both examples of strongly central series. | The [[lower central series]] and [[upper central series]] of a [[nilpotent group]] are both examples of strongly central series. {{further|[[Lower central series is strongly central]], [[Upper central series is strongly central]]}} | ||
==Relation with other properties== | ==Relation with other properties== | ||
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===Weaker properties=== | ===Weaker properties=== | ||
* [[Normal series]] | * [[Stronger than::Normal series]]: {{proofat|[[Strongly central series implies normal series]]}} | ||
* [[Central series]] | * [[Stronger than::Central series]]: {{proofat|[[Strongly central series implies central series]]}} | ||
==References== | |||
===Textbook references=== | |||
* {{booklink-defined|KhukhroNGA}}, Page 76, Section 3.2 (formal definition, around equation 3.2.5) | |||
Latest revision as of 16:04, 30 June 2008
This article defines a property that can be evaluated for a subgroup series
Definition
is termed strongly central if for every .
(A similar definition works for transfinite series).
Examples
The lower central series and upper central series of a nilpotent group are both examples of strongly central series. Further information: Lower central series is strongly central, Upper central series is strongly central
Relation with other properties
Weaker properties
- Normal series: For full proof, refer: Strongly central series implies normal series
- Central series: For full proof, refer: Strongly central series implies central series
References
Textbook references
- Nilpotent groups and their automorphisms by Evgenii I. Khukhro, ISBN 3110136724, More info, Page 76, Section 3.2 (formal definition, around equation 3.2.5)