Groupprops, The Group Properties Wiki (pre-alpha)

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Contents 1 Statement 2 Related facts 3 Facts used 4 Proof

Statement

Let G be a nilpotent group and let c be the nilpotency class of G.Then, G is a solvable group, and if l denotes the derived length of G, we have:

.

Related facts

• Nilpotent implies solvable
• Derived length gives no upper bound on nilpotency class: Knowing the solvable length of a nilpotent group gives no bound on its nilpotence class. In fact, there are metabelian groups of arbitrarily large class, such as the dihedral 2-groups and the generalized quaternion groups.

Facts used

1. Second half of lower central series of nilpotent group comprises abelian groups: If G is nilpotent of class c, and G_k denotes the k^th term of the lower central series of G, then G_k is Abelian for .

Proof

We prove this by induction on the nilpotence class. Note that the statement is true when c = 1 or c = 2.

Given: A finite nilpotent group G of class c.

To prove: The solvable length of G is at most log₂c + 1.

Proof: Let k be the smallest positive integer greater than or equal to (c + 1) / 2. In other words, either k = (c + 1) / 2 or k = c / 2 + 1, depending on the parity of c. Then, G_k is an Abelian group, and G / G_k is a group of class k − 1, which is at most c / 2.

By the induction assumption, we have:

.

Thus, G has an Abelian normal subgroup such that the solvable length of the quotient is at most log₂c. This yields that the solvable length of G is at most log₂c + 1.