Abelian series

Definition
A subgroup series of a group is termed an abelian series if both these conditions hold:


 * The series is a subnormal series.
 * Each quotient group between successive members of the series is an abelian group.

A group possesses an abelian series of finite length if and only if it is a solvable group.

Explicitly, in symbols, consider a series:

$$G = K_0 \ge K_1 \ge \dots \ge K_n = 1$$

The series is abelian if each $$K_{i+1}$$ is a normal subgroup of $$K_i$$ and each quotient group $$K_i/K_{i+1}$$ is an abelian group.