Penultimate term of lower central series is abelian in nilpotent group of class at least three
From Groupprops
Contents
Statement
If is a Nilpotent group (?) and the Nilpotence class (?) of is an integer , then the term of the Lower central series (?) of is Abelian. In particular, it is an Abelian characteristic subgroup of .
Related facts
Corollaries
- Finite nilpotent and every Abelian characteristic subgroup is central implies class at most two
- Nilpotence class is strictly greater than solvable length in finite nilpotent group of class at least three
Breakdown of corresponding statement for upper central series
The second term of the upper central series of a finite nilpotent group need not be abelian. Further information: Second term of upper central series not is abelian, Upper central series may be tight with respect to nilpotence class
Facts used
- Second half of lower central series of nilpotent group comprises Abelian groups: If is a group of nilpotence class , and , the member of the lower central series of is Abelian.
Proof
The proof follows directly from fact (1), and the observation that for , .
References
Textbook references
- Finite Groups by Daniel Gorenstein, ISBN 0821843427, ^{More info}, Page 214, Exercise 3, Chapter 5