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From Groupprops

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

Contents 1 Definition 1.1 Symbol-free definition 2 Facts 2.1 Relation with nilpotence class

Definition

Symbol-free definition

Given a solvable group, we define its 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.

Facts

Relation with nilpotence class

Further information: Nilpotence 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.