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At first sight, finitely generated abelian groups and canonical forms of matrices appear to have little in common. However, reduction to Smith normal form, named after its originator H. Smith in , is a matrix version of the Euclidean algorithm and is exactly what the theory requires in both cases. Under certain conditions each matrix is similar to a diagonal or nearly diagonal matrix, namely its Jordan form. The reader is assumed to be familiar with the elementary properties of rings and fields.

Also a knowledge of abstract linear algebra including vector spaces, linear mappings, matrices, bases and dimension is essential, although much of the theory is covered in the text but from a more general standpoint: the role of vector spaces is widened to modules over commutative rings. The fundamental theorem of finitely generated abelian groups can be stated two ways, generalizing the two forms of the fundamental theorem of finite abelian groups.

Undergraduate Mathematics/Abelian group

The theorem, in both forms, in turn generalizes to the structure theorem for finitely generated modules over a principal ideal domain , which in turn admits further generalizations. The primary decomposition formulation states that every finitely generated abelian group G is isomorphic to a direct sum of primary cyclic groups and infinite cyclic groups.

A primary cyclic group is one whose order is a power of a prime. That is, every finitely generated abelian group is isomorphic to a group of the form. Again, the rank n and the invariant factors k 1 , The rank and the sequence of invariant factors determine the group up to isomorphism. The history and credit for the fundamental theorem is complicated by the fact that it was proven when group theory was not well-established, and thus early forms, while essentially the modern result and proof, are often stated for a specific case.

GT22. The Fundamental Theorem of Finite Abelian Groups

As far as the fundamental theorem on finite abelian groups is concerned, it is not clear how far back in time one needs to go to trace its origin. The fundamental theorem for finite abelian groups was proven by Leopold Kronecker in Kronecker , using a group-theoretic proof, [4] though without stating it in group-theoretic terms; [5] a modern presentation of Kronecker's proof is given in Stillwell , 5. This generalized an earlier result of Carl Friedrich Gauss from Disquisitiones Arithmeticae , which classified quadratic forms; Kronecker cited this result of Gauss's.

Finitely Generated Abelian Groups

The theorem was stated and proved in the language of groups by Ferdinand Georg Frobenius and Ludwig Stickelberger in The fundamental theorem for finitely presented abelian groups was proven by Henry John Stephen Smith in Smith , [3] as integer matrices correspond to finite presentations of abelian groups this generalizes to finitely presented modules over a principal ideal domain , and Smith normal form corresponds to classifying finitely presented abelian groups.

There is the additional technicality of showing that a finitely presented abelian group is in fact finitely generated , so Smith's classification is not a complete proof for finitely generated abelian groups. This was done in the context of computing the homology of a complex, specifically the Betti number and torsion coefficients of a dimension of the complex, where the Betti number corresponds to the rank of the free part, and the torsion coefficients correspond to the torsion part. If this is , then for some , so can be the relabelled.

If it is more, then we must have at least two nonzero — reorder so that they are and , with. So by induction, there is a generating set with.

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  • Clearly this is the generating set we want. Now go back to our original situation and find the generating set. We now have , so is the first member of a -element generating set and has order , which is smaller than the order of , which is a contradiction since is minimal. So, in fact,. By induction,. But each either has infinite order or order. So is respectively , or.

    And so, reordering, we have:. There are actually different ways to do this: in , for relatively prime, the element has order , and the group has elements, so generates the whole group, and we have. So a cyclic group of order with prime factorization is isomorphic to. In particular, a finitely generated abelian group can be written as a direct sum of a free subgroup and a finite set of cyclic groups of prime power order.

    Finitely Generated Abelian Groups and Similarity of Matrices over a Field |

    There is only one way to do this, as can be seen by counting elements: if there are elements of order , then occurs in of the factors, and so on. We call the the elementary divisors of.

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    Alternatively, starting with an elementary divisor decomposition, we could let where there is one copy of each distinct prime from among the invariant factors, with its highest exponent. Then we could let be the product of all the distinct prime powers remaining, again with their highest exponents, and so on.

    If the elementary divisors are then we have.

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    The also have the property that divides for each. We call them the invariant factors — since the are unique, so are the. Both of these decompositions are occasionally useful, especially for finite abelian groups. There are very many of these, and at a certain stage it becomes hard to tell the difference. For example, and are both non-finitely-generated, and includes as a subgroup, but what exactly is the difference? In […].