# Abelian groups

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### Definition

An abelian group is a structure G = (G, + ,−,0), where + is an infix binary operation, called the group addition, is a prefix unary operation, called the group negative and 0 is a constant (nullary operation), called the additive identity element, such that
+ is commutative:   x + y = y + x,
+ is associative:   (x + y) + z = x + (y + z),
0 is an additive identity for + :   0 + x = x, and
gives an additive inverse for + :   x + x = 0.

### Morphisms

Let G and H be abelian groups. A morphism from G to H is a function h : GH that is a homomorphism: h(x + y) = h(x) + h(y).

Remark: It follows that h(−x) = −h(x)  and  h(0) = 0.

### Examples

(Z, + , −, 0), the integers, with addition, unary subtraction, and zero. The variety of abelian groups is generated by this algebra.

### Properties

 Classtype variety Equational theory decidable in polynomial time Quasiequational theory decidable First-order theory decidable [W. Szmielew, Decision problem in group theory, Library of the Tenth International Congress of Philosophy, Amsterdam, August 11--18, 1948, Vol.1, Proceedings of the Congress (1949) 763--766 MRreview] Locally finite no Residual size ω Congruence distributive no (Z2×Z2) Congruence modular yes Congruence n-permutable yes, n = 2, p(x,y,z) = x−y + z Congruence regular yes, congruences are determined by subalgebras Congruence uniform yes Congruence types permutational Congruence extension property yes, if K ≤ H ≤ G then K ≤ G Definable principal congruences no Equationally definable principal congruences no Amalgamation property yes Strong amalgamation property yes Epimorphisms are surjective yes

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### Subclasses

Boolean groups
Commutative rings

### Superclasses

Groups
Commutative monoids

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