# Tour:Abelian group

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PREREQUISITES: Definition of group. Return to group if you don't remember this
WHAT YOU NEED TO DO:
• Read, and thoroughly understand, the definition of Abelian group given below. If this is confusing, you might want to return to the page on group and revisit the definition
• Understand the notation and conventions for an Abelian group

## 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.

## 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.

## Examples

VIEW: groups satisfying this property | groups dissatisfying this property
VIEW: Related group property satisfactions | Related group property dissatisfactions

### 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).

PONDER (WILL BE EXPLORED LATER IN THE TEXT):
• Can you think of an example of a non-Abelian group? How are those examples different from the abelian groups?
• How do you use Abelianness for many of the Abelian groups you usually deal with (like integers, rational numbers etc.) How would the lack of Abelianness affect things?
WHAT'S BELOW: Some more examples of abelian groups and general information regarding abelian groups. Go through it. It may use terminology and ideas that you haven't encountered; ignore those parts.
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### 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.