Composition length

This article defines an arithmetic function on a restricted class of groups, namely: group of finite composition lengths

Definition

The composition length of a group is defined as the length of any composition series of the group. The composition length is well-defined only for a group of finite composition length.

Suppose $G$ is a group with a composition series:

$\{e\}=N_{0}\leq N_{1}\leq N_{2}\leq \dots \leq N_{r}=G$ .

Then, the composition length of $G$ is defined to be equal to $r$ .

Every finite group has finite composition length.
The composition length of a solvable group is the sum of the exponents of all prime factors of its order. This is because all its composition factors are cyclic groups of prime order.
A group has composition length zero if and only if it is the trivial group.
A group has composition length one if and only if it is a simple group.