more precise results about -torsion. For example, if for a prime then the -primary component of the stable homotopy group vanishes unless is divisible by , in which case it is cyclic of order . The J-homomorphism An important subgroup of , for , is the image of the J-homomorphism , where denotes the special orthogonal group. In the stable range , the homotopy groups only depend on . This period 8 pattern is known as Bott periodicity, and it is reflected in the stable homotopy groups of spheres via the image of the -homomorphism which is: a cyclic group of order 2 if is congruent to 0 or 1 modulo 8; trivial if is congruent to 2, 4, 5, or 6 modulo 8; and a cyclic group of order equal to the denominator of , where is a Bernoulli number, if . This last case accounts for the elements of unusually large finite order in for such values of . For example, the stable groups have a cyclic subgroup of order 504, the denominator of . The stable homotopy groups of spheres are the direct sum of the image of the -homomorphism, and the kernel of the Adams -invariant, a homomorphism from these groups to . Roughly speaking, the image of the -homomorphism is the subgroup of "well understood" or "easy" elements of the stable homotopy groups. These well understood elements account for most elements of the stable homotopy groups of spheres in small dimensions. The quotient of by the image of the -homomorphism is considered to be the "hard" part of the stable homotopy groups of spheres . (Adams also introduced certain order 2 elements of for , and these are also considered to be "well understood".) Tables of homotopy groups of spheres sometimes omit the "easy" part to save space. Ring structure The direct sum of the stable homotopy groups of spheres is a supercommutative graded ring, where multiplication is given by composition of representing maps, and any element of non-zero degree is nilpotent; the nilpotence theorem on complex cobordism implies Nishida's theorem. Example: If is the generator of (of order 2), then is nonzero and generates , and is nonzero and 12 times a generator of , while is zero because the group is trivial. If and and are elements of with and , there is a Toda bracket of these elements. The Toda bracket is