Free class two group

Definition
A free class two group on a set $$S$$ is defined in the following equivalent ways:


 * It is the quotient $$F(S)/(F(S),F(S)],F(S)])$, where $F(S)$ is the free group on the set $S$. Note that $[[F(S),F(S)],F(S)]$ can be verbally described as the third member of the [[lower central series of $$F(S)$$.
 * It is the free algebra on $$S$$ in the variety of groups of nilpotency class two.

The isomorphism type of $$F(S)$$ depends only on the cardinality of $$S$$.

Note that any free class two group is in particular a reduced free group.