Orbits of the automorphism group of a module over a principal ideal ring

We describe the orbits of the automorphism group of a finitely generated module over a principal ideal domain in terms of canonical representatives and by a complete system of invariants. For a primary module, we establish a natural bijection between the set of orbits and the set of partitions of the Young diagram corresponding to the module into two Young diagrams, which allows us to determine the number of orbits.