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.
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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 $a_{1}+a_{2}+\dots +a_{n}$.

Related facts

• Composition series of direct product is obtained by piecing together composition series of direct factors
• Composition length of direct product is sum of composition lengths
• Composition length of extension group is sum of composition lengths
• Chief length of extension group is bounded by sum of chief lengths