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
VIEW: Definitions built on this | Facts about this: (facts closely related to Abelian series, all facts related to Abelian series) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
View a complete list of semi-basic definitions on this wiki

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.