Abelian series

This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definition
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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:

$G = K_{0} \geq K_{1} \geq \dots \geq K_{n} = 1$

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.