Simplicial surfaces with given automorphism group

Abstract For any finite group G , Frucht presents a construction of a cubic graph whose automorphism group is isomorphic to G . For groups generated by two elements, we simplify Frucht’s construction to obtain cubic graphs with fewer nodes. In the general case, we address an oversight in Frucht’s construction. We prove the existence of cycle double covers of the resulting cubic graphs that lead to simplicial surfaces with given finite automorphism group. For almost all finite non-abelian simple groups, we give alternative constructions based on graphic regular representations. In the general cases $$C_n,D_n,A_5$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>D</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>5</mml:mn> </mml:msub> </mml:mrow> </mml:math> for $$n\ge 4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> , we provide alternative constructions of simplicial spheres. Furthermore, we embed these simplicial spheres into the Euclidean 3-Space with equilateral triangles such that the automorphism group of the simplicial spheres and the symmetry group of the corresponding polyhedron are isomorphic.