Finitely generated and parafree not implies free

This article gives the statement and possibly, proof, of a non-implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., finitely generated parafree group) need not satisfy the second group property (i.e., free group)
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Statement

It is possible to have a finitely generated parafree group that is not a finitely generated free group (and hence, not a free group).

In fact, the example we construct will be a finitely presented group.

Proof

Suppose $i,j$ are integers such that <amth>ij \ne 0</math> (i.e., neither of them is zero). Consider the group:

$H_{i,j}=\langle a,b,c\mid a=[c^{i},a][c^{j},b]\rangle$

The group $H_{i,j}$ is parafree. Explicitly, it has the same lower central series quotient groups as free group:F2.