Derived length

This article defines an arithmetic function on a restricted class of groups, namely: solvable groups

Definition

Symbol-free definition

Given a solvable group, we define its derived length or solvable length as follows:

• It is the length of the derived series of the group. Note here that by length of the series, we mean the number of successive inclusions, so the length is one less than the actual number of subgroups in the derived series.
• It is the minimum possible length of a subnormal series from the trivial subgroup to the whole group such that all the quotients in the series are abelian groups.

When we say that a group has derived length $k$, we typically mean that it has solvable length at most $k$.

Facts

Small values

• A group has derived length zero if and only if it is trivial.
• A group has derived length at most one if and only if it is abelian.
• A group has derived length at most two if and only if it is a metabelian group: it has an abelian normal subgroup such that the quotient group is also an abelian group.

Relation with nilpotency class

Further information: Nilpotency class versus solvable length

Any nilpotent group is solvable, and there are numerical relations between the nilpotence class and solvable length:

• Solvable length is logarithmically bounded by nilpotence class
• Solvable length gives no upper bound on nilpotence class: For a solvable length greater than $1$, the value of the solvable length gives no upper bound on the value of the nilpotence class.