# Definition:Length of Group

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## Definition

Let $G$ be a finite group.

The **length** of $G$ is the length of a composition series for $G$.

That is, the **length** of $G$ is the number of factors in a composition series for $G$ (not including $G$ itself).

The **length** of $G$ can be denoted $\map l G$.

By the Jordan-Hölder Theorem, all composition series for $G$ have the same length.

Therefore, the **length** of a finite group $G$ is well-defined.

## Sources

- 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $2$: Normal and Composition Series: $\S 73 \beta$