# Chief length of direct product is sum of chief lengths

From Groupprops

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: (factscloselyrelated to chief length, all facts related to chief length)

View facts about external direct product: (factscloselyrelated to external direct product, all facts related to external direct product)

View facts about sum: (factscloselyrelated to sum, all facts related to sum)

## Statement

### For two groups

Suppose and are groups with chief lengths and . Then, the chief length of the external direct product is the sum .

In particular, if both and are groups of finite chief length, then so is . Conversely, if is a group of finite chief length, then so are and .

### For multiple groups

Suppose are groups with chief lengths . Then, the chief length of the external direct product is the sum .

## 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