In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after early 19th century mathematician Niels Henrik Abel.^[1]

The concept of an abelian group underlies many fundamental algebraic structures, such as fields, rings, vector spaces, and algebras. The theory of abelian groups is generally simpler than that of their non-abelian counterparts, and finite abelian groups are very well understood and fully classified.

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Definition

An abelian group is a set ${\displaystyle A}$, together with an operation ${\displaystyle \cdot }$ that combines any two elements ${\displaystyle a}$ and ${\displaystyle b}$ of ${\displaystyle A}$ to form another element of ${\displaystyle A,}$ denoted ${\displaystyle a\cdot b}$. The symbol ${\displaystyle \cdot }$ is a general placeholder for a concretely given operation. To qualify as an abelian group, the set and operation, ${\displaystyle (A,\cdot )}$, must satisfy four requirements known as the abelian group axioms (some authors include in the axioms some properties that belong to the definition of an operation: namely that the operation is defined for any ordered pair of elements of A, that the result is well-defined, and that the result belongs to A):

Associativity

For all ${\displaystyle a}$, ${\displaystyle b}$, and ${\displaystyle c}$ in ${\displaystyle A}$, the equation ${\displaystyle (a\cdot b)\cdot c=a\cdot (b\cdot c)}$ holds.

Identity element

There exists an element ${\displaystyle e}$ in ${\displaystyle A}$, such that for all elements ${\displaystyle a}$ in ${\displaystyle A}$, the equation ${\displaystyle e\cdot a=a\cdot e=a}$ holds.

Inverse element

For each ${\displaystyle a}$ in ${\displaystyle A}$ there exists an element ${\displaystyle b}$ in ${\displaystyle A}$ such that ${\displaystyle a\cdot b=b\cdot a=e}$, where ${\displaystyle e}$ is the identity element.

Commutativity

For all ${\displaystyle a}$, ${\displaystyle b}$ in ${\displaystyle A}$, ${\displaystyle a\cdot b=b\cdot a}$.

A group in which the group operation is not commutative is called a "non-abelian group" or "non-commutative group".^[2]: 11

Facts

Notation

There are two main notational conventions for abelian groups – additive and multiplicative.

Convention	Operation	Identity	Powers	Inverse
Addition	${\displaystyle x+y}$	0	${\displaystyle nx}$	${\displaystyle -x}$
Multiplication	${\displaystyle x\cdot y}$ or ${\displaystyle xy}$	1	${\displaystyle x^{n}}$	${\displaystyle x^{-1}}$

Generally, the multiplicative notation is the usual notation for groups, while the additive notation is the usual notation for modules and rings. The additive notation may also be used to emphasize that a particular group is abelian, whenever both abelian and non-abelian groups are considered, some notable exceptions being near-rings and partially ordered groups, where an operation is written additively even when non-abelian.^{[3]: 28–29 [4]: 9–14}

Multiplication table

To verify that a finite group is abelian, a table (matrix) – known as a Cayley table – can be constructed in a similar fashion to a multiplication table.^[5]: 10 If the group is ${\displaystyle G=\{{}_{}}$Zdroj:https://en.wikipedia.org?pojem=Abelian_group
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