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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 ...
Since the group of integers Z serves as a generator, the category Ab is therefore a Grothendieck category; indeed it is the prototypical example of a Grothendieck category. 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.
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 ...
Subobjects and quotient objects are well-behaved in abelian categories. For example, the poset of subobjects of any given object A is a bounded lattice. Every abelian category A is a module over the monoidal category of finitely generated abelian groups; that is, we can form a tensor product of a finitely generated abelian group G and any ...
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 ...
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.
p-groups of the same order are not necessarily isomorphic; for example, the cyclic group C 4 and the Klein four-group V 4 are both 2-groups of order 4, but they are not isomorphic. Nor need a p-group be abelian; the dihedral group Dih 4 of order 8 is a non-abelian 2-group. However, every group of order p 2 is abelian. [note 1]
An Abelian 2-group is a groupoid (that is, a category in which every morphism is an isomorphism) with a bifunctor +: and natural transformations: + +: (+) + + (+) which satisfy a host of axioms ensuring these transformations behave similarly to commutativity and associativity () for an Abelian group.