# Composition length of extension group is sum of composition lengths

This article gives an expression for the value of the arithmetic function composition length of a group obtained by applying a group operation group extension in terms of the values for the input groups. It says that the value for the group obtained after performing the operation is the sum of the values for the input groups.
View facts about composition length: (facts closely related to composition length, all facts related to composition length)
View facts about group extension: (facts closely related to group extension, all facts related to group extension)
View facts about sum: (facts closely related to sum, all facts related to sum)

## Definition

Suppose $G$ is a group, $N$ is a normal subgroup, and $G/N$ is the corresponding quotient group.

The composition length of $G$ is the sum of the composition length of $N$ and the composition length of $G/N$.

In particular, if $G$ is a group of finite composition length, then so are $N$ and $G/N$. Conversely, if both $N$ and $G/N$ have finite composition length, so does $G$.