Discrete Infinite Groups

A group will be known as the infinite group if it contains an infinite number of elements. In a group, we can define the infinite set of elements with the help of some constructive process or formula. This type of formula may have some parameters that can be real numbers, integer numbers or even points of manifold. In the informal classification, the beginning points are described as follows:

In the first case, the groups will be known as discrete and in the second case, they will be known as continuous. The infinite cyclic group is the best and simple example of a discrete infinite group. The group (I, +) is an example of the infinite group because Set I contains infinite sets of integers.

Suppose there is a group G and a set X. With the help of function φ : G × X → X, we can define the group action of G on set X. This function satisfies some conditions for all x ∈ X, which are described as follows:

• φ(e, x) = x, where e is used to indicate the identity element of G.
• φ(g, φ(h, x)) = φ(gh, x) for all g, h ∈ G

There are some infinite groups (rational or integers) which do not contain continuous groups. The infinite groups contain those types of elements which do not parameterize with the help of continuous parameters like complex numbers or real numbers. There are some points that show how we can form a group in the case of rational numbers, non-rational numbers and integer numbe rs, which are described as follows:

1. Groups can be formed by the integer Z and rational numbers Q under the addition operation. In this case, the identity will be zero, and the inverse of a number q will be -q.

2. Groups can be formed by the non-zero rational numbers Q* under the multiplication operation.

In this case, the inverse of non-zero rational q will be the non-zero rational 1/q, and the identity will be one.

3. SL(2, Z).

Here we will show the example of a discrete infinite group in the form of string theory with the help of a group of all two by two matrices of unit determinants which have integer entries.

The group operation is a type of matrix multiplication.

Explicitly, the following formula is used to show the group multiplication:

Here all the above described entries are integers.

In the above equation, the unit determinant condition can be described by the condition ad - bc = 1.

When there are complex numbers, in this case, the group acts naturally. Here are complex numbers x + iy with y>0.

Now we will use a complex number z = x+iy where x and y both are used to indicate the real numbers and y>0 and then find the action of matrix on this complex number like this:

We will get the group GL(2, Z) if we only need that the determinant ad - bc be non-zero.

4. SL(2, Z, n)

This example is the same as above described in the SL(2, Z) part. The only difference is that in the group, we only need the top right entry of each matrix to be the multiple of n. Here n is used to indicate a fixed positive integer.