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 $G$ together with a function

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

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, $\forall x,y,z\in G:(xy)z=x(yz)$
2. For the given law of composition there exists a unique left identity, that is there exists a unique $e\in G$ such that $\forall g\in G:eg=g$.
3. For each $g\in G$, there exists an inverse of $g$, that is an element of $G$ denoted $g^{-1}$ 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 $8\cdot 10^{53}$ 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,

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

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

$x+y$ instead of $xy$.

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 $\det {\begin{pmatrix}a&b\\c&d\end{pmatrix}}=ad-bc$ 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: