Chief length

Definition

Let $G$ be a group. The chief length of $G$ is defined to be the length of a chief series for $G$, that 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:

$G=N_{0}\geq N_{1}\geq \dots \geq N_{r}=\{e\}$

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:

$G=N_{0}\geq N_{1}\geq \dots \geq N_{r}=\{e\}$

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 $G/N_{i}$.

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, so this is well-defined.

Facts

• For $G$ a p-group, the chief length is equal to the logarithm of the order of the group with that prime as the base. For example, a group of order $16$ has chief length $4$, since $16=2^{4}$.
• Chief length of direct product is sum of chief lengths