Chief length of direct product is sum of chief lengths

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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.

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