Derived length gives no upper bound on nilpotency class
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Contents |
Statement
For l > 1, there exist nilpotent groups of solvable length l and arbitrarily large nilpotence class.
Related facts
Converse
Proof
Dihedral groups
We first show that for l = 2, there exist groups of arbitrarily large nilpotence class.
For
, the dihedral group
, given by the presentation:
,
has nilpotence class n − 1, but solvable length 2, since it has an abelian normal subgroup
such that the quotient is also an abelian group.
To get an example of a group of length exactly l for l > 2 that has arbitrarily large nilpotence class, take the direct product of
with any nilpotent group of solvable length precisely l.