Abelian group

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History

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.

Definition

Symbol-free definition

An Abelian group is a group where any two elements commute.

Definition with 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 $G$).

Examples

Cyclic groups are good examples of Abelian groups. 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.

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.

Facts

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 normal with Abelian quotient group if and only if the subgroup contains the commutator subgroup.

Metaproperties

Varietal group property

This group property is a varietal group property, in the sense that the collection of groups satisfying this property forms a variety of algebras. In other words, the collection of groups satisfying this property is closed under taking subgroups, taking quotients and taking arbitrary direct products.

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

Subgroups

This group property is subgroup-closed, viz., any subgroup of a group satisfying the property also satisfies the property
View a complete list of subgroup-closed group properties

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. For full proof, refer: Abelianness is subgroup-closed

Quotients

This group property is quotient-closed, viz., any quotient of a group satisfying the property also has the property
View a complete list of quotient-closed group properties

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. For full proof, refer: Abelianness is quotient-closed

Direct products

This group property is direct product-closed, viz., the direct product of an arbitrary (possibly infinite) family of groups each having the property, also has the property
View other direct product-closed group properties

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. For full proof, refer: Abelianness is direct product-closed

Testing

The testing problem

Further information: Abelianness testing problem

GAP command

This group property can be tested using built-in functionality of Groups, Algorithms, Programming (GAP).
View GAP-testable group properties

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.

Study of this notion

Mathematical subject classification

Under the Mathematical subject classification, the study of this notion comes under the class: 20K

References

Textbook references

• Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347, More info, Page 17 (definition as Point (2) in general definition of a group)
• Groups and representations by Jonathan Lazare Alperin and Rowen B. Bell, ISBN 0387945261, More info, Page 2 (definition introduced in paragraph)
• Topics in Algebra by I. N. Herstein, More info, Page 28 (formal definition)
• A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613, More info, Page 2 (formal definition)
• Finite Group Theory (Cambridge Studies in Advanced Mathematics) by Michael Aschbacher, ISBN 0521786754, More info, Page 1 (definition introduced in paragraph)