Nilpotency of fixed class is direct product-closed
Version in terms of fixed class bound
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 .