Strongly central series

This article defines a property that can be evaluated for a subgroup series

View a complete list of properties of subgroup series

Definition

A subgroup series

$G=K_{1}\geq K_{2}\geq K_{3}\geq \ldots \geq K_{c}\geq K_{c+1}=1$

is termed strongly central if $[K_{i},K_{j}]\leq K_{i+j}$ for every $i,j=1,2,\ldots ,c$.

(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)