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==Definition==

===Symbol-free definition===

An abelian group is a group in which any two elements commute.

===Full definition===
An '''abelian group''' is a set <math>G</math> equipped with a (infix) binary operation <math>+</math> (called the addition or group operation), an identity element <math>0</math> and a (prefix) unary operation <math>-</math>, called the inverse map or negation map, satisfying the following:

* For any <math>a,b,c \in G</math>, <math>a + (b + c) = (a + b) + c</math>. This property is termed [[associativity]].
* For any <math>a \in G</math>, <math>a + 0 = 0 + a = a</math>. <math>0</math> thus plays the role of an additive [[identity element]] or [[neutral element]].
* For any <math>a \in G</math>, <math>a + (-a) = (-a) + a = 0</math>. Thus, <math>-a</math> is an [[inverse element]] to <math>a</math> with respect to <math>+</math>.
* For any <math>a,b \in G</math>, <math>a + b = b + a</math>. This property is termed [[commutativity]].

==Notation==

When <math>G</math> is an abelian group, we typically use ''additive'' notation and terminology. Thus, the group multiplication is termed ''addition'' and the product of two elements is termed the ''sum''.

# The infix operator <math>+</math> is used for the group multiplication, so the sum of two elements <math>a</math> and <math>b</math> is denoted by <math>a + b</math>. The group multiplication is termed ''addition'' and the product of two elements is termed the ''sum''.
# The identity element is typically denoted as <math>0</math> and termed ''zero''
# The inverse of an element is termed its ''negative'' or ''additive inverse''. The inverse of <math>a</math> is denoted <math>-a</math>.
# <math>a + a + \ldots + a</math> done <math>n</math> times is denoted <math>na</math>, (where <math>n \in \mathbb{N}</math>) while <math>(-a) + (-a) + (-a) + \ldots + (-a)</math> done <math>n</math> times is denoted <math>(-n)a</math>.

This convention is typically followed in a situation where we are dealing with the abelian group <math>G</math> in isolation, rather than as a subgroup of a possibly non-abelian group. If we are working with subgroups in a non-abelian group, we typically use multiplicative notation even if the subgroup happens to be abelian.

==Examples==

===Some infinite examples===

The additive group of integers <math>\mathbb{Z}</math>, the additive group of rational numbers <math>\mathbb{Q}</math>, the additive group of real numbers <math>\mathbb{R}</math>, the multiplicative group of nonzero rationals <math>\mathbb{Q}^*</math>, and the multiplicative group of nonzero real numbers <math>\mathbb{R}^*</math> are some examples of Abelian groups.

(More generally, for any field, the additive group, and the multiplicative group of nonzero elements, are abelian groups).

===Finite examples===

[[Cyclic group]]s are good examples of abelian groups, where the cyclic group of order <math>n</math> is the group of integers modulo <math>n</math>.

===Additive groups of rings===

The additive group of the ring of integers, and more generally of any ring, is an abelian group.

===Arithmetic functions under Dirichlet product===

{{further|[[Multiplicative functions form abelian group under Dirichlet product]]}}

The [[arithmetic function]]s, i.e., functions from the set of natural numbers to a [[commutative unital ring]], admit a commutative and associative binary operation called the [[Dirichlet product]]. Those arithmetic functions whose value at <math>1</math> is invertible are invertible with respect to this Dirichlet product.

The multiplicative functions form a subgroup of this group.

===Arithmetic functions under pointwise addition===

The arithmetic functions from the natural numbers to a commutative unital ring form an abelian group under pointwise addition.

Abelian group - Revision history

Vipul: Created page with '==Definition== ===Symbol-free definition=== An abelian group is a group in which any two elements commute. ===Full definition=== An '''abelian group''' is a set G...'

Vipul: Created page with '==Definition== ===Symbol-free definition=== An abelian group is a group in which any two elements commute. ===Full definition=== An '''abelian group''' is a set $G$ ...'