# Composition length of direct product is sum of composition lengths

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.
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## Statement

### For two groups

Suppose $G_1$ and $G_2$ are groups with composition lengths $a_1$ and $a_2$. Then, the composition length of the external direct product $G_1 \times G_2$ is the sum $a_1 + a_2$.

In particular, if both $G_1$ and $G_2$ are groups of finite composition length, then so is $G_1 \times G_2$. Conversely, if $G_1 \times G_2$ is a group of finite composition length, then so are $G_1$ and $G_2$.

### For multiple groups

Suppose $G_1,G_2,\dots,G_n$ are groups with composition lengths $a_1,a_2,\dots,a_n$. Then, the composition length of the external direct product $G_1 \times G_2 \times \dots \times G_n$ is the sum $a_1 + a_2 + \dots + a_n$.