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.

Particular cases

Size of $S$	Isomorphism type of free class two group on $S$
0	trivial group
1	group of integers
2	unitriangular matrix group:UT(3,Z)