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

Related facts

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