Abelian group

An Abelian group is a group whose operation is commutative: combining any two elements gives the same result in either order.

Group (mathematics): A set with an associative operation, an identity element, and an inverse for every element. An Abelian group is precisely a group that also satisfies commutativity.
Cyclic group: A group generated by repeated application of one element and its inverse. Every cyclic group is Abelian, making it a basic example of the structure.
Integers under addition: The group of integers with addition, zero as identity, and additive inverses. This infinite cyclic group is the standard example of an Abelian group.
Fundamental theorem of finite Abelian groups: A classification theorem expressing every finite Abelian group as a product of cyclic groups of prime-power order. It gives a complete structural classification of finite cases.
Non-Abelian group: A group whose operation is not commutative for at least one pair of elements. It is the direct alternative: some element pairs give different results when reversed.
Binary operation: A rule that assigns an element of a set to each ordered pair of elements from that set. The operation is the rule whose order-independence defines Abelian groups.
Direct product of groups: A group formed from tuples of elements of groups, with operations performed coordinatewise. Products of Abelian groups remain Abelian, coordinate by coordinate.
Finite cyclic group: A cyclic group with finitely many elements, generated by one element of finite order. Its operation is commutative because every element is a power of the same generator.
Character group: A group of homomorphisms from a group to a multiplicative group of scalars. Characters translate Abelian group structure into scalar-valued functions.
Commutator subgroup: The subgroup generated by all commutators of a group. It measures noncommutativity and is trivial exactly when the group is Abelian.

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