Abelians

In mathematics, "Abelians" is most commonly encountered as the plural adjective referring to concepts and structures related to abelian groups. An abelian group, also called a commutative group, is a group in which the group operation is commutative. More formally, a group (G, *) is said to be abelian if for all elements a and b in G, the equation a * b = b * a holds. The term "abelian" honors the mathematician Niels Henrik Abel.

The concept of abelian groups is fundamental in abstract algebra and has applications across various areas of mathematics, including number theory, algebraic topology, and algebraic geometry. Many important mathematical structures, such as the integers under addition, form abelian groups.

Properties of abelian groups are often simpler and more manageable than those of non-abelian groups. For instance, subgroups and quotient groups of abelian groups are always abelian. Additionally, finitely generated abelian groups have a well-understood structure, as they can be expressed as a direct sum of cyclic groups.

The term "Abelians" might also be used less frequently to refer to:

• Abelian functions: A type of complex function of several variables with multiple periods.
• Related Mathematical Concepts: Adjectives derived from "Abel" or "Abelian" appearing in other mathematical contexts.