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.