Abstract Algebra/Definition of groups, very basic properties

DefinitionsEdit

The following definition is the starting point of group theory.

Definition 1.1:

A group is a set   together with a function

 

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,  
  2. For the given law of composition there exists a unique left identity, that is there exists a unique   such that  .
  3. For each  , there exists an inverse of  , that is an element of   denoted   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   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,

 .

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

  instead of  .

ExamplesEdit

Example 1.3:

A classical example of a group are the invertible   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   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   (that is,  ), and the binary operation is given by the only choice we have:

 .

This construct satisfies all the group axioms.

Elementary propertiesEdit

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


ExponentiationEdit

If   is a group,   an element and  , we can raise   to the  -th power. This works as follows: