# Composition length of direct product is sum of composition lengths

From Groupprops

This article gives an expression for the value of the arithmetic function composition length of an external direct product in terms of the values for the direct factors. It says that the value for the direct product is the sum of the values for the direct factors.

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

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

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

## Statement

### For two groups

Suppose and are groups with composition lengths and . Then, the composition length of the external direct product is the sum .

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

### For multiple groups

Suppose are groups with composition lengths . Then, the composition length of the external direct product is the sum .