# Abelian Groups

### Some articles on abelian groups, group, abelian, groups, abelian group:

Category Of Abelian Groups
... In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms ... This is the prototype of an abelian category ... The monomorphisms in Ab are the injective group homomorphisms, the epimorphisms are the surjective group homomorphisms, and the isomorphisms are the bijective group homomorphisms ...
Torsion-free Abelian Groups Of Rank 1
... Infinitely generated abelian groups have very complex structure and are far less well understood than finitely generated abelian groups ... Even torsion-free abelian groups are vastly more varied in their characteristics than vector spaces ... Torsion-free abelian groups of rank 1 are far more amenable than those of higher rank, and a satisfactory classification exists, even though there are an uncountable number of isomorphism classes ...
List Of First-order Theories - Groups
... The signature of group theory has one constant 1 (the identity), one function of arity 1 (the inverse) whose value on t is denoted by t−1, and one function of arity 2 ... Groups are defined by the axioms Identity ∀x 1x = x ∧ x1 = x Inverse ∀x x−1x = 1 ∧ xx−1 = 1 Associative ∀x∀y∀z (xy)z = x(yz) Some properties of groups that can be defined in the ... any fixed positive integer n) The theory of Abelian groups is decidable ...
Yoneda Lemma - Preadditive Categories, Rings and Modules
... A preadditive category is a category where the morphism sets form abelian groups and the composition of morphisms is bilinear examples are categories of ... functors from the original category into the category of abelian groups these are functors which are compatible with the addition of morphisms and should be thought of as forming a module category over the ... — in fact, the enlarged version is an abelian category, a much more powerful condition ...
Indecomposable Module - Examples - PID
... that this is simple if and only if n=1 (or p=0) for example, the cyclic group of order 4, Z/4, is indecomposable but not simple – it has the subgroup 2Z/4 of order 2, but this does not have a complement ... Over the integers Z, modules are abelian groups ... A finitely-generated abelian group is indecomposable if and only if it is isomorphic to Z or to a factor group of the form for some prime number p and some positive integer n ...

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