Given a group \(\mathcal {G}\) , let \(\alpha (\mathcal {G})\) (resp. \(\alpha ^{P}(\mathcal {G})\) ) denote the minimal number of vertices among all graphs (resp. planar graphs) \(\Gamma \) such that \(\mathop {\textrm{Aut}}\Gamma \cong \mathcal {G}\) . Over the years, several researchers have constructed vertex-minimal planar graphs with cyclic group symmetry. For an abelian group \(\mathcal {G}\) , we construct planar graphs whose automorphism group is isomorphic to \(\mathcal {G}\) . Consequently, we obtain suitable upper bounds for \(\alpha ^P(\mathcal {G})\) with equability holding for a large class of abelian groups. This partially addresses one of the open questions raised by Archer et al. (J Algebr Combin 54:1–15, 2021): find the value of \(\alpha ^{P}(\mathcal {G})\) when \(\mathcal {G}\) is an abelian group. Further, we classify all finite abelian groups \(\mathcal {G}\) such that \(\alpha (\mathcal {G})=\alpha ^{P}(\mathcal {G})\) , giving a partial answer to another question asked by Archer et al.
