 Solution

Let $1=A_0\le \cdots \le A_r=A$ and $1=B_0\le \cdots \le B_s=B$ be central series. Pad the shorter chain by repeating its terminal group, so both have length $t=\max (r,s)$. In the direct product of groups, group commutators are coordinatewise, henceThe equality holds because commutators supported in either coordinate generate the corresponding factor independently. Thereforeis a central series, proving that a direct product of two nilpotent groups is nilpotent. More precisely, the coordinatewise recurrence for the lower central series gives ${\Gamma }_j(A\times B)={\Gamma }_j(A)\times {\Gamma }_j(B)$, so the nilpotency class of a direct product is the maximum of the two factor classes.