Each finite group is a symmetry group of some map (an “Atom”-bifurcation)

Maps are studied, i.e. cell decompositions of closed two-dimensional surfaces, or two-dimensional atoms, which encode bifurcations of Liouville fibrations of nondegenerate integrable Hamiltonian systems. Any finite group $G$ is proved to be the symmetry group of an orientable map (of an atom). Moreover one such a map $X(G)$ is constructed algorithmically. Upper bounds are obtained for the minimal genus M$g(G)$ of an orientable map with the given symmetry group $G,$ and for the minimal number of vertices, edges and sides of such maps.