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

we used the fact that $$\det\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc$$ 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: