# Composition length of semidirect product is sum of composition lengths

From Groupprops

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: (factscloselyrelated to composition length, all facts related to composition length)

View facts about external semidirect product: (factscloselyrelated to external semidirect product, all facts related to external semidirect product)

View facts about sum: (factscloselyrelated to sum, all facts related to sum)

## Contents

## Statement

### For external semidirect product

Suppose and are groups, and gives an action of on by automorphisms. Suppose the composition length of is and the composition length of is . Then the composition length of the external semidirect product is .

### For internal semidirect product

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

Suppose the composition length of is and the composition length of is . Then the composition length of the external semidirect product is .