Abelian group

In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that

Properties

If n is a natural number and x is an element of an abelian group G, then nx can be defined as x + x + ... + x (n summands) and (−n)x = −(nx). In this way, G becomes a module over the ring Z of integers. In fact, the modules over Z can be identified with the abelian groups.

Related Topics:
Natural number - Module - Ring

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Theorems about abelian groups (i.e. modules over the principal ideal domain Z) can often be generalized to theorems about modules over an arbitrary principal ideal domain. A typical example is the classification of finitely generated abelian groups.

Related Topics:
Principal ideal domain - Finitely generated abelian group

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If f, g : G  →  H are two group homomorphisms between abelian groups, then their sum f + g, defined by (f + g)(x) = f(x) + g(x), is again a homomorphism. (This is not true if H is a non-abelian group). The set Hom(G, H) of all group homomorphisms from G to H thus turns into an abelian group in its own right.

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Somewhat akin to the dimension of vector spaces, every abelian group has a rank. It is defined as the cardinality of the largest set of linearly independent elements of the group. The integers and the rational numbers have rank one, as well as every subgroup of the rationals. While the rank one torsion-free abelian groups are well understood, even finite-rank abelian groups are not well understood. Infinite-rank abelian groups can be extremely complex and many open questions exist, often intimately connected to questions of set theory.

Related Topics:
Dimension - Vector space - Rank - Cardinality - Linearly independent - Rational number - Torsion - Set theory

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