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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. 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 fundamental theorem of finite abelian groups states that every finite abelian group can be expressed as the direct sum of cyclic subgroups of prime-power order; it is also known as the basis theorem for finite abelian groups. Moreover, automorphism groups of cyclic groups are examples of abelian groups. [13]
The main structure theorem about this group is the Mordell–Weil theorem which shows this group is in fact a finitely-generated abelian group. Moreover, there are many conjectures related to this group, such as the Birch and Swinnerton-Dyer conjecture which relates the rank of A ( K ) {\displaystyle A(K)} to the zero of the associated L ...
Every elementary abelian p-group is a vector space over the prime field with p elements, and conversely every such vector space is an elementary abelian group. By the classification of finitely generated abelian groups, or by the fact that every vector space has a basis, every finite elementary abelian group must be of the form (Z/pZ) n for n a ...
For example, if G is any group, then there exists an automorphism σ of G × G that switches the two factors, i.e. σ(g 1, g 2) = (g 2, g 1). For another example, the automorphism group of Z × Z is GL(2, Z), the group of all 2 × 2 matrices with integer entries and determinant, ±1. This automorphism group is infinite, but only finitely many ...
completely determine its homology groups with coefficients in A, for any abelian group A: (,) Here might be the simplicial homology, or more generally the singular homology. The usual proof of this result is a pure piece of homological algebra about chain complexes of free abelian groups.
The second Prüfer theorem states that a countable abelian p-group whose non-trivial elements have finite p-height is isomorphic to a direct sum of cyclic groups. Examples show that the assumption that the group be countable cannot be removed. The two Prüfer theorems follow from a general criterion of decomposability of an abelian group into a ...
An abelian group A is torsion-free if and only if it is flat as a Z-module, which means that whenever C is a subgroup of some abelian group B, then the natural map from the tensor product C ⊗ A to B ⊗ A is injective. Tensoring an abelian group A with Q (or any divisible group) kills torsion. That is, if T is a torsion group then T ⊗ Q = 0.