Chief series

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 defining ingredient::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.

Weaker properties

 * Stronger than::Normal series

Related properties

 * Composition series
 * Subnormal series