# Solvability of fixed length is direct product-closed

From Groupprops

## Contents

## Statement

### Version in terms of fixed class bound

Suppose is a collection of groups indexed by an indexing set . Suppose there is a positive integer such that each is a solvable group of derived length at most .

Then, the external direct product of the s is also a solvable group of derived length at most .

### Version in terms of maximum class

Suppose is a collection of groups indexed by an indexing set . If all the s are solvable groups and there is a common finite bound on their derived length values, then the external direct product of the s is also a solvable group and its derived length is the maximum of the derived length values of all the s.

In particular, for two solvable groups and of derived lengths respectively, the derived length of equals .