# Chief length of direct product is sum of chief lengths

This article gives an expression for the value of the arithmetic function chief 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 chief length: (facts closely related to chief length, all facts related to chief length)
View facts about external direct product: (facts closely related to external direct product, all facts related to external direct product)
View facts about sum: (facts closely related to sum, all facts related to sum)

## Statement

### For two groups

Suppose $G_1$ and $G_2$ are groups with chief lengths $a_1$ and $a_2$. Then, the chief 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 chief length, then so is $G_1 \times G_2$. Conversely, if $G_1 \times G_2$ is a group of finite chief length, then so are $G_1$ and $G_2$.

### For multiple groups

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