  Below we construct non-cyclic and torsion-free abelian quotients for
subgroups of braid groups generated by cube powers of half-twists. In the case
of 3 and 4 strands we compute the abelianization of these groups. Also, we get
abelianizations of level-3 congruence subgroups of braid groups on 3 and 4
strands. Moreover, we construct non-cyclic and torsion-free abelian quotients
for level-3 congruence subgroups of braid groups. The constructions are based
on the original construction of Dennis Johnson \cite{DJ}, who computed an
abelian quotient of the Torelli group.
