Nilpotent product

Definition
Suppose $$c$$ is a positive integer, and $$G_i, i \in I$$ are groups. The class $$c$$ nilpotent product of the groups $$G_i$$ is defined in a number of equivalent ways.


 * 1) As a varietal product: It is the defining ingredient::varietal product of the groups $$G_i$$ with respect to the subvariety of nilpotent groups of nilpotency class $$\le c$$ in the variety of groups.
 * 2) Explicit definition: It is the quotient of the external free product of the $$G_i$$s by the normal closure in it of the join of all subgroups obtained as iterated commutators of length $$c+1$$ or more between the images of the $$G_i$$s and that involve at least two of the $$G_i$$s.

The case of two groups
We consider here the case of two groups $$G_1$$ and $$G_2$$. For simplicity of notation, we identify the groups $$G_i$$ with their images in the nilpotent product.