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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 ...
Given any pair A, B of objects in an abelian category, there is a special zero morphism from A to B. This can be defined as the zero element of the hom-set Hom(A,B), since this is an abelian group. Alternatively, it can be defined as the unique composition A → 0 → B, where 0 is the zero object of the abelian category.
Pages in category "Abelian group theory" The following 37 pages are in this category, out of 37 total. This list may not reflect recent changes. 0–9. Abelian 2 ...
In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods ...
In 1870, Leopold Kronecker gave a definition of an abelian group in the context of ideal class groups of a number field, generalizing Gauss's work. [25] Ernst Kummer's attempts to prove Fermat's Last Theorem resulted in work introducing groups describing factorization into prime numbers. [26]
Abelian category, in category theory, a preabelian category in which every monomorphism is a kernel and every epimorphism is a cokernel; Abelian and Tauberian theorems, in real analysis, used in the summation of divergent series; Abelian extension, in Galois theory, a field extension for which the associated Galois group is abelian
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 ...
An object in Ab is injective if and only if it is a divisible group; it is projective if and only if it is a free abelian group. The category has a projective generator (Z) and an injective cogenerator (Q/Z). Given two abelian groups A and B, their tensor product A⊗B is defined; it is again an abelian group.