# Chief series

This article defines a property that can be evaluated for a subgroup seriesView a complete list of properties of subgroup series

## Contents

## Definition

### Definition for finite length

Let be a group. A chief series for is a subgroup series from to the trivial subgroup where all members are normal subgroups of , and where the series cannot be refined further. More explicitly:

- A series of subgroups:

is termed a chief series if are normal in for all , is a proper subgroup of , and there is no normal subgroup of that properly contains and is properly contained within . In other words, the normal series cannot be refined further to another normal series.

- A series of subgroups:

is termed a chief series if each is normal in and is a minimal normal subgroup of .

A group that possesses a chief series is termed a group of finite chief length. The factor groups for a chief series are termed the chief factors. It turns out that for a group of finite chief length, any two chief series have the same length and the lists of chief factors are the same.