Abstract Algebra/Definition of groups, very basic properties

Definitions

The following definition is the starting point of group theory.

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 $8\cdot 10^{53}$ 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:

Examples

Example 1.3:

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

Failed to parse (unknown function "\middle"): {\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 $\det \left(\begin{array}{cc}a& b\\ c& d\end{array}\right)=ad-bc$ and a matrix is invertible if and only if its determinant is not zero.

Example 1.4:

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

$ee:=e$.

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