# Nilpotency of fixed class is direct product-closed

From Groupprops

(Redirected from Nilpotency class of direct product is maximum of nilpotency classes)

## 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 nilpotent group of nilpotency class at most .

Then, the external direct product of the s is also a nilpotent group of nilpotency class at most .

### Version in terms of maximum class

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

In particular, for two nilpotent groups and of nilpotency classes respectively, the nilpotency class of equals .