In the same paper, Serre found the first place that -torsion occurs in the homotopy groups of dimensional spheres, by showing that has no -torsion if , and has a unique subgroup of order if and . The case of 2-dimensional spheres is slightly different: the first -torsion occurs for . In the case of odd torsion there are more precise results; in this case there is a big difference between odd and even dimensional spheres. If is an odd prime and , then elements of the -component of have order at most . This is in some sense the best possible result, as these groups are known to have elements of this order for some values of . Furthermore, the stable range can be extended in this case: if is odd then the double suspension from to is an isomorphism of -components if , and an epimorphism if equality holds . The -torsion of the intermediate group can be strictly larger.