Composition length

Symbol-free definition
The composition length of a group is defined as the length of any defining ingredient::composition series of the group. The composition length is well-defined only for a defining ingredient::group of finite composition length.

By the Jordan-Holder theorem, the composition length is well-defined, because all composition series have the same length and in fact the same collection of composition factors.

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

$$\{ e \} = N_0 \le N_1 \le N_2 \le \dots \le N_r = G$$.

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