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An abelian group is a set, together with an operation ･ , that combines any two elements and of to form another element of , denoted .The symbol ･ is a general placeholder for a concretely given operation.
If a free abelian group is a quotient of two groups /, then is the direct sum /. [4] Given an arbitrary abelian group , there always exists a free abelian group and a surjective group homomorphism from to .
The Klein four-group is the smallest non-cyclic group. It is, however, an abelian group, and isomorphic to the dihedral group of order (cardinality) 4, symbolized (or , using the geometric convention); other than the group of order 2, it is the only dihedral group that is abelian.
Abelian varieties appear naturally as Jacobian varieties (the connected components of zero in Picard varieties) and Albanese varieties of other algebraic varieties. The group law of an abelian variety is necessarily commutative and the variety is non-singular. An elliptic curve is an abelian variety of dimension 1.
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 ...
The Cayley table tells us whether a group is abelian. Because the group operation of an abelian group is commutative, a group is abelian if and only if its Cayley table's values are symmetric along its diagonal axis. The group {1, −1} above and the cyclic group of order 3 under ordinary multiplication are both examples of abelian groups, and ...
A group with () = {} for some n in N is called a solvable group; this is weaker than abelian, which is the case n = 1. A group with () {} for all n in N is called a non-solvable group. A group with () = {} for some ordinal number, possibly infinite, is called a hypoabelian group; this is weaker than solvable, which is the case α is finite (a ...
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.