do this, one uses the fact that if splits as a direct sum of subgroups of coprime order, then Given this, the fundamental theorem shows that to compute the automorphism group of it suffices to compute the automorphism groups of the Sylow -subgroups separately (that is, all direct sums of cyclic subgroups, each with order a power of ). Fix a prime and suppose the exponents of the cyclic factors of the Sylow -subgroup are arranged in increasing order: for some . One needs to find the automorphisms of One special case is when , so that there is only one cyclic prime-power factor in the Sylow -subgroup . In this case the theory of automorphisms of a finite cyclic group can be used. Another special case is when is arbitrary but for . Here, one is considering to be of the form so elements of this subgroup can be viewed as comprising a vector space of dimension over the finite field of elements . The automorphisms of this subgroup are therefore given by the invertible linear transformations, so where is the appropriate general linear group. This is easily shown to have order In the most general case, where the and are arbitrary, the automorphism group is more difficult to determine. It is known, however, that if one defines and then one has in particular , , and One can check that this yields the orders in the previous examples as special cases (see Hillar, C., & Rhea, D.). Finitely generated abelian groups An abelian group is finitely generated if it contains a finite set of elements (called generators) such that every element of the group is a linear combination with integer coefficients of elements of . Let be a free abelian group with basis There is a unique group homomorphism such that This homomorphism is surjective, and its kernel is finitely generated (since integers form a Noetherian ring). Consider the matrix with integer entries, such that the entries of its th column are the coefficients of the th generator of the kernel. Then, the abelian group is isomorphic to the cokernel of linear map defined by . Conversely every integer matrix defines a finitely generated abelian group. It follows that the study of finitely generated abelian groups is totally equivalent with the study of integer matrices. In particular, changing the generating set of is equivalent with multiplying on the