# Length of Subgroup Plus Length of Quotient Group

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## Theorem

Let $G$ be a finite group.

Let $H$ be a normal subgroup of $G$.

Then:

- $\map l G = \map l H + \map l {G / H}$

where:

- $\map l G$ denotes the length of $G$
- $G / H$ denotes the quotient group of $G$ by $H$.

## Proof

## Sources

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