# 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 $\max \{ c_1, c_2 \}$.