# Abstract Algebra/Definition of groups, very basic properties

## Definitions

The following definition is the starting point of group theory.

Definition 1.1:

A group is a set ${\displaystyle G}$ together with a function

${\displaystyle G\times G\to G,(x,y)\mapsto xy}$

called multiplication or binary operation and denoted simply by juxtaposition of that group, such that the following rules hold:

1. The law of composition is associative, that is, ${\displaystyle \forall x,y,z\in G:(xy)z=x(yz)}$
2. For the given law of composition there exists a unique left identity, that is there exists a unique ${\displaystyle e\in G}$ such that ${\displaystyle \forall g\in G:eg=g}$.
3. For each ${\displaystyle g\in G}$, there exists an inverse of ${\displaystyle g}$, that is an element of ${\displaystyle G}$ denoted ${\displaystyle g^{-1}}$ such that.

Although these axioms to be satisfied by a group are quite brief, groups may be very complex, and the study of groups is not trivial. For instance, there exists a very complicated group, called the Monster group, which has roughly ${\displaystyle 8\cdot 10^{53}}$ elements and the law of composition is so complicated that even modern computers have difficulty doing computations in this group.

There is a special type of groups (namely those that are commutative, i.e. the multiplication obeys the commutative law), which are named after the famous mathematician Niels Henrik Abel:

Definition 1.2:

An Abelian group is a group such that its binary operation is commutative, that is,

${\displaystyle \forall x,y\in G:xy=yx}$.

Oftentimes, Abelian groups are written additively, that is, for the binary operation of ${\displaystyle x}$ and ${\displaystyle y}$ we write

${\displaystyle x+y}$ instead of ${\displaystyle xy}$.

## Examples

Example 1.3:

A classical example of a group are the invertible ${\displaystyle 2\times 2}$ matrices with real entries. Formally, this group can be written down like this:

$\displaystyle GL_2(\mathbb R) := \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \middle| a, b, c, d \in \mathbb R, ad - bc \neq 0 \right\}$ ;

we used the fact that ${\displaystyle \det {\begin{pmatrix}a&b\\c&d\end{pmatrix}}=ad-bc}$ and a matrix is invertible if and only if its determinant vanishes.

Example 1.4:

The trivial group is the group which contains only one element, call it ${\displaystyle e}$ (that is, ${\displaystyle G=\{e\}}$), and the binary operation is given by the only choice we have:

${\displaystyle ee:=e}$.

This construct satisfies all the group axioms.

## Elementary properties

Here we describe properties that all groups share, which are immediate consequences of the definition 1.1.

## Exponentiation

If ${\displaystyle G}$ is a group, ${\displaystyle g\in G}$ an element and ${\displaystyle k\in \mathbb {Z} }$, we can raise ${\displaystyle g}$ to the ${\displaystyle k}$-th power. This works as follows: