# Abelian group

Abelian groups generalize the arithmetic concept of addition of integers. (They were named after Niels Henrik Abel.[1]).

• Thus, in abstract algebra, an abelian group, is a commutative group, previously called a `symplectic' group.

An abelian group is defined as a set, ${\displaystyle A}$, together with an operation "${\displaystyle *}$" which is commutative, that is, for any elements ${\displaystyle x}$ and ${\displaystyle y}$ of ${\displaystyle A}$ one has that:
${\displaystyle x*y=y*x.}$

The operation of addition of integers obviously has the property of commutativity.

## Notes

• 1. The collection of all Abelian groups, together with the group homomorphisms between them, forms the category ${\displaystyle Ab}$, the prototype of an Abelian category.

## References

1. Jacobson (2009), p. 41
2. http://aux.planetphysics.us/files/papers/100/NvalLogicsGG.pdf Georgescu, G. 2006, N-valued Logics and Łukasiewicz-Moisil Algebras, Axiomathes, 16 (1-2): 123-136.
3. http://images.planetphysics.us/files/lec/294/CategoryOfLMnLogicAlgebrasPP.pdf Topic: Algebraic category of Łukasiewicz-Moisil n-valued logic algebras.
• Jacobson, Nathan (2009). Basic Algebra I (2nd ed.). Dover Publications. ISBN 978-0-486-47189-1.
• Szmielew, Wanda (1955). "Elementary properties of abelian groups". Fundamenta Mathematicae 41: 203–271.
• Cox, David (2004). Galois Theory. Wiley-Interscience. MR 2119052.
• Fuchs, László (1970). Infinite Abelian Groups, Vol. I. Pure and Applied Mathematics. 36-I. Academic Press. MR 0255673.