Abelian group

Origin of the term
The term abelian group comes from Niels Henrick Abel, a mathematician who worked with groups even before the formal theory was laid down, in order to prove unsolvability of the quintic.

The word abelian is usually begun with a small a.

Definition
An abelian group is a group where any two elements commute. In symbols, a group $$G$$ is termed abelian if for any elements $$x$$ and $$y$$ in $$G$$, $$xy = yx$$ (here $$xy$$ denotes the product of $$x$$ and $$y$$ in $$G$$). Note that $$x,y$$ are allowed to be equal, though equal elements commute anyway, so we can restrict attention if we wish to unequal elements.

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


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

Equivalent formulations

 * A group is abelian if its defining ingredient::center is the whole group.
 * A group is abelian if its defining ingredient::derived subgroup is trivial.

Notation
When $$G$$ 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.


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

This convention is typically followed in a situation where we are dealing with the abelian group $$G$$ 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.

Some infinite examples
The additive group of integers $$\mathbb{Z}$$, the additive group of rational numbers $$\mathbb{Q}$$, the additive group of real numbers $$\mathbb{R}$$, the multiplicative group of nonzero rationals $$\mathbb{Q}^*$$, and the multiplicative group of nonzero real numbers $$\mathbb{R}^*$$ 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 groups are good examples of abelian groups, where the cyclic group of order $$n$$ is the group of integers modulo $$n$$.

Further, any direct product of cyclic groups is also an abelian group. Further, every finitely generated Abelian group is obtained this way. This is the famous structure theorem for finitely generated abelian groups.

The structure theorem can be used to generate a complete listing of finite abelian groups, as described here: classification of finite Abelian groups.

Non-examples
Not every group is abelian. The smallest non-abelian group is the symmetric group on three letters: the group of all permutations on three letters, under composition. Its being non-abelian hinges on the fact that the order in which permutations are performed, matters.

Occurrence as subgroups
Every cyclic group is abelian. Since each group is generated by its cyclic subgroups, every group is generated by a family of abelian subgroups. A trickier question is: do there exist abelian normal subgroups? A good candidate for an abelian normal subgroup is the center, which is the collection of elements of the group that commute with every element of the group.

Occurrence as quotients
The maximal abelian quotient of any group is termed its abelianization, and this is the quotient by the commutator subgroup. A subgroup is an abelian-quotient subgroup (i.e., normal with abelian quotient group) if and only if the subgroup contains the commutator subgroup.

Formalisms
A group $$G$$ is an abelian group if and only if, in the external direct product $$G \times G$$, the diagonal subgroup $$\{ (g,g) \mid g \in G \}$$ is a normal subgroup.

Metaproperties
Abelian groups form a variety of algebras. The defining equations for this variety are the equations for a group along with the commutativity equation.

Any subgroup of an abelian group is abelian -- viz., the property of being abelian is subgroup-closed. This follows as a direct consequence of abelianness being varietal.

Any quotient of an abelian group is abelian -- viz the property of being abelian is quotient-closed. This again follows as a direct consequence of abelianness being varietal.

A direct product of abelian groups is abelian -- viz the property of being abelian is direct product-closed. This again follows as a direct consequence of abelianness being varietal.

The testing problem
To test whether a group is abelian, the GAP syntax is:

IsAbelian (group)

where group either defines the group or gives the name to a group previously defined.

Textbook references

 * , Page 17 (definition as Point (2) in general definition of a group)
 * , Page 2 (definition introduced in paragraph)
 * , Page 42 (defined immediately after the definition of group, as a group where the composition is commutative)
 * , Page 28 (formal definition)
 * , Page 2 (formal definition)
 * , Page 1 (definition introduced in paragraph)