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Abelian groups of rank 0 are exactly the periodic abelian groups. The group Q of rational numbers has rank 1. Torsion-free abelian groups of rank 1 are realized as subgroups of Q and there is a satisfactory classification of them up to isomorphism. By contrast, there is no satisfactory classification of torsion-free abelian groups of rank 2. [2 ...
The rank of a symmetry group is closely related to the complexity of the object (a molecule, a crystal structure) being under the action of the group. If G is a crystallographic point group, then rank(G) is up to 3. [9] If G is a wallpaper group, then rank(G) = 2 to 4. The only wallpaper-group type of rank 4 is p2mm. [10]
To qualify as an abelian group, the set and operation, (,), must satisfy four requirements known as the abelian group axioms (some authors include in the axioms some properties that belong to the definition of an operation: namely that the operation is defined for any ordered pair of elements of A, that the result is well-defined, and that the ...
is an abelian group NS(V), called the Néron–Severi group of V. This is a finitely-generated abelian group by the Néron–Severi theorem, which was proved by Severi over the complex numbers and by Néron over more general fields. In other words, the Picard group fits into an exact sequence
An important step in the proof of the classification of finitely generated abelian groups is that every such torsion-free group is isomorphic to a . A non-finitely generated countable example is given by the additive group of the polynomial ring Z [ X ] {\displaystyle \mathbb {Z} [X]} (the free abelian group of countable rank).
A pre-abelian category is an additive category with all kernels and cokernels. An abelian category is a pre-abelian category such that every monomorphism and epimorphism is normal. The preadditive categories most commonly studied are in fact abelian categories; for example, Ab is an abelian category.
Pages in category "Abelian group theory" The following 37 pages are in this category, out of 37 total. ... Rank of an abelian group; S. Slender group; T.
Torsion-free abelian group, an abelian group which is a torsion-free group; Torsion-free rank, the cardinality of a maximal linearly independent subset of an abelian group or of a module over an integral domain