Chief series

This article defines a property that can be evaluated for a subgroup series

View a complete list of properties of subgroup series

Definition

Definition for finite length

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

• A series of subgroups: $G = N_0 \ge N_1 \ge \dots \ge N_r = \{ e \}$

is termed a chief series if $N_i$ are normal in $G$ for all $i$, $N_{i+1}$ is a proper subgroup of $N_i$, and there is no normal subgroup of $G$ that properly contains $N_{i+1}$ and is properly contained within $N_i$. In other words, the normal series cannot be refined further to another normal series.

• A series of subgroups: $G = N_0 \ge N_1 \ge \dots \ge N_r = \{ e \}$

is termed a chief series if each $N_i$ is normal in $G$ and $N_{i-1}/N_i$ is a minimal normal subgroup of $G/N_i$.

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.