Free fixed-class group

Definition
Suppose $$c$$ is a nonnegative integer and $$S$$ is a set. The free group of nilpotency class $$c$$ on $$S$$ is defined in the following equivalent ways:


 * 1) It is the quotient of $$F(S)$$, the free group on $$S$$, by the $$(c+1)^{th}$$ member of the lower central series of $$F(S)$$.
 * 2) It is the free algebra on $$S$$ in the variety of groups of nilpotency class $$c$$ (this variety includes all groups whose nilpotency class is less than or equal to $$c$$.