Groupprops, The Group Properties Wiki (pre-alpha)

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Contents 1 Statement 2 Related facts 2.1 Converse 3 Proof 3.1 Dihedral groups

Statement

For l > 1, there exist nilpotent groups of solvable length l and arbitrarily large nilpotence class.

Related facts

• Solvable not implies nilpotent

Converse

• Nilpotent implies solvable
• Solvable length is logarithmically bounded by nilpotence class

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.