  Given a finite group $G$, the {\it genus spetrum} ${\rm sp}(G)$ of $G$ is the
set of integers $g\geq 0$ such that $G$ can act faithfully on an orientable
closed surface of genus $g$ by orientation-preserving homeomorphisms. The
determination of ${\rm sp}(G)$ is a classical topic and has a long history, but
progress is lacked. In this paper, when $G$ is an abelian $p$-group with $p>2$,
we propose a new approach to ${\rm sp}(G)$, giving a structural description for
${\rm sp}(G)$ in terms of a function which can be computed in finitely many
steps.
