# Composition length of extension group 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 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: (factscloselyrelated to composition length, all facts related to composition length)

View facts about group extension: (factscloselyrelated to group extension, all facts related to group extension)

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

## Definition

Suppose is a group, is a normal subgroup, and is the corresponding quotient group.

The composition length of is the sum of the composition length of and the composition length of .

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