N-solvable group

This group property is natural number-parametrized, in other words, for every natural number, we get a corresponding group property

Definition

Suppose $G$ is a group and $n$ is an integer. We say that $G$ is $n$-solvable if the n-derived series of $G$ reaches the trivial subgroup in finitely many steps. The number of steps it takes to do so is termed the n-derived length of $G$.

Terminology caution

Note that the notion of n-solvable group is distinct from the notion of p-solvable group. In particular, when somebody talks of $n$-solvability for a prime number $n$, it is more likely that they are referring to it in the sense of p-solvability rather than n-solvability as defined here.