A new method for derivation of symmetry groups of non-rigid molecules is presented, based on the concept of isometry of nuclear configurations defined by a certain number of internal (structural) coordinates. The substitution group of the internal coordinates interrelating isometric nuclear configurations generates the internal isometric group by both permutations and intransitive imprimitive substitution groups, and together with the covering group it generates the full isometric group. Examples for these groups are given. It is shown that the rotation-internal motion hamiltonian is symmetric with respect to the isometric group. The relation between various approaches to the isometric group is investigated; the role of automorphisms is pointed out. The new approach is shown to give a self-consistent procedure for the quantum mechanical problem associated with the rotation-internal motion hamiltonian.