On separating sets of polynomial invariants of finite abelian group actions

Let $G$ be a finite group acting on a finite dimensional complex vector space $V$ via linear transformations. Let $\mathbb{C}[V{]}^G$ be the algebra of polynomials that are invariant under the induced $G$-action on the polynomial ring $\mathbb{C}[V]$. A subset $S\subseteq \mathbb{C}[V{]}^G$ is a separating set if it separates the orbits of the group action. If $G$ is abelian, then there exist finite separating sets consisting of monomials. In this paper we investigate properties of separating sets from four different points of view, including the monoid theoretical properties of separating sets consisting of monomials, the minimal size of separating sets consisting of monomials, the exact value of the separating Noether number $\text{sepbeta}(G)$ of abelian groups of rank $4$, and the inverse problem of $\text{sepbeta}(G)$ for abelian groups of rank $2$.