The derived $\pi$-length, nilpotent $\pi$-length, and simple $\pi$-length of finite $\pi$-soluble groups

The concept of the derived $\pi$-length for finite $\pi$-soluble groups is introduced and its elementary properties are described. The dependence between the $\pi$-length, nilpotent $\pi$-length, and derived $\pi$-length, and also between the derived and nilpotent lengths of a $\pi$-Hall subgroup is determined.