  Garoufalidis and Levine introduced the homology cobordism group of homology
cylinders over a surface. This group can be regarded as a generalization of the
mapping class group. Using torsion invariants, we show that the abelianization
of this group is infinitely generated provided that the first Betti number of
the surface is positive. In particular, this shows that the group is not
perfect. This answers questions of Garoufalidis-Levine and Goda-Sakasai.
Furthermore we show that the abelianization of the group has infinite rank for
the case that the surface has more than one boundary component. These results
hold for the homology cylinder analogue of the Torelli group as well.
