# Composition length of semidirect product 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 external semidirect product 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 external semidirect product: (facts closely related to external semidirect product, all facts related to external semidirect product)
View facts about sum: (facts closely related to sum, all facts related to sum)

## Statement

### For external semidirect product

Suppose $N$ and $H$ are groups, and $\rho:H \to \operatorname{Aut}(N)$ gives an action of $H$ on $N$ by automorphisms. Suppose the composition length of $N$ is $a$ and the composition length of $H$ is $b$. Then the composition length of the external semidirect product $N \rtimes H$ is $a + b$.

### For internal semidirect product

Suppose $G$ is a group, $N$ is a normal subgroup and $H$ is a complement to $N$ in $G$. In other words, $G$ is an internal semidirect product of $N$ and $H$.

Suppose the composition length of $N$ is $a$ and the composition length of $H$ is $b$. Then the composition length of the external semidirect product $N \rtimes H$ is $a + b$.