  If $\Gamma$ is a finitely generated Fuchsian group such that its derived
subgroup $\Gamma'$ is co-compact and torsion free, then $S={\mathbb
H}^{2}/\Gamma'$ is a closed Riemann surface of genus $g \geq 2$ admitting the
abelian group $A=\Gamma/\Gamma'$ as a group of conformal automorphisms. We say
that $A$ is a homology group of $S$. A natural question is if $S$ admits unique
homology groups or not, in other words, is there are different Fuchsian groups
$\Gamma_{1}$ and $\Gamma_{2}$ with $\Gamma_{1}'=\Gamma'_{2}$? It is known that
if $\Gamma_{1}$ and $\Gamma_{2}$ are both of the same signature
$(0;k,\ldots,k)$, for some $k \geq 2$, then the equality
$\Gamma_{1}'=\Gamma_{2}'$ ensures that $\Gamma_{1}=\Gamma_{2}$. Generalizing
this, we observe that if $\Gamma_{j}$ has signature $(0;k_{j},\ldots,k_{j})$
and $\Gamma_{1}'=\Gamma'_{2}$, then $\Gamma_{1}=\Gamma_{2}$. We also provide
examples of surfaces $S$ with different homology groups. A description of the
normalizer in ${\rm Aut}(S)$ of each homology group $A$ is also obtained.
