The Separating Noether Number of Finite Abelian Groups

For a finite abelian group $G$, let ${\beta }_{\mathrm{s}\mathrm{e}\mathrm{p}}(G)$ denote its separating Noether number. We determine ${\beta }_{\mathrm{s}\mathrm{e}\mathrm{p}}(G)$ exactly for every finite abelian group $G\cong C_{n_1}\oplus \cdots \oplus C_{n_r}$ with $1<n_1\mid \cdots \mid n_r\mathrm{.}$ If $r=2s-1$, then $${\beta }_{\mathrm{s}\mathrm{e}\mathrm{p}}(G)=n_s+n_{s+1}+\cdots +n_r,$$ whereas if $r=2s$, then $${\beta }_{\mathrm{s}\mathrm{e}\mathrm{p}}(G)=\frac{n_s}{p_1}+n_{s+1}+\cdots +n_r,$$ where $p_1$ denotes the smallest prime divisor of $n_1$. Our proof is additive-combinatorial in nature. It avoids the Davenport-equality assumption $\mathsf{D}(n_{s}G)={\mathsf{D}}^{\ast }(n_{s}G)$ 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 $\mathrm{\mid }{G}_0\mathrm{\mid }\le r+1$ satisfies $\mathrm{\mid }\text{supp}(A)\mathrm{\mid }=\mathrm{\mid }{G}_0\mathrm{\mid }=r+1$. 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.