The Separating Noether Number of Finite Abelian Groups

For a finite abelian group $G$, let $\beta_{\mathrm{sep}}(G)$ denote its separating Noether number. We determine $\beta_{\mathrm{sep}}(G)$ exactly for every finite abelian group $G \cong C_{n_1}\oplus \cdots \oplus C_{n_r}$ with $1<n_1 \mid \cdots \mid n_r.$ If $r=2s-1$, then $$\beta_{\mathrm{sep}}(G)=n_s+n_{s+1}+\cdots+n_r,$$ whereas if $r=2s$, then $$\beta_{\mathrm{sep}}(G)=\frac{n_s}{p_1}+n_{s+1}+\cdots+n_r,$$ where $p_1$ denotes the smallest prime divisor of $n_1$. Our proof is additive-combinatorial in nature. It avoids the Davenport-equality assumption $\mathsf{D}(n_sG)=\mathsf{D}^{*}(n_sG)$ used in previous works. The key ingredients are a geometric reduction of auxiliary sequences via the novel construction of geodesic surrogates, alongside a uniform lifting procedure for relation groups. As an application, we prove that if $r\ge 2$, then every extremal separating atom $A$ over $G_0$ with $|G_0|\le r+1$ satisfies $|\supp(A)|=|G_0|=r+1$. Equivalently, the conjectured support conclusion of Schefler, Zhao, and Zhong holds for all finite abelian groups of rank at least $2$. By contrast, the rank-$1$ case is exceptional: for cyclic groups, the analogous conjectural conclusion is false.