Nilpotency of fixed class is direct product-closed

Statement

Version in terms of fixed class bound

Suppose $G_{i}, i \in I$ is a collection of groups indexed by an indexing set $I$ . Suppose there is a positive integer $c$ such that each $G_{i}$ is a nilpotent group of nilpotency class at most $c$ .

Then, the external direct product of the $G_{i}$ s is also a nilpotent group of nilpotency class at most $c$ .

Version in terms of maximum class

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

In particular, for two nilpotent groups $G_{1}$ and $G_{2}$ of nilpotency classes $c_{1}, c_{2}$ respectively, the nilpotency class of $G_{1} \times G_{2}$ equals $max {c_{1}, c_{2}}$ .

Statement

Version in terms of fixed class bound

Version in terms of maximum class

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