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Section 1.5 Presentation

If you have constructed the multiplication table for \(S_3\) as an exercise, you have probably come to the realization that writing down multiplication tables becomes quite annoying rather quickly for higher order groups. There has to be a better way of encoding the abstract group structure of finite groups without having to write down the multiplication table each time. One such way is to write down the “presentation” of a group.

The idea is to specify a set of generators \(S\) of the group, which is a subset of elements of the group from which all other elements can be obtained by group multiplication, and the essential relations \(R\) that the generators satisfy. We write the presentation as \(\langle S | R \rangle\text{.}\) Without getting into too much detail, let me simply give a few examples of presentations.

As a basic example, consider the group \(\mathbb{Z}_2\) of two elements. Its presentation would be given as:

\begin{equation*} \langle a | a^2 = e \rangle. \end{equation*}

Similarly, the cyclic group with \(n\) elements \(\mathbb{Z}_n\) would be written as:

\begin{equation*} \langle a | a^n = e \rangle. \end{equation*}

The direct product \(\mathbb{Z}_2 \times \mathbb{Z}_2\) would be written as:

\begin{equation*} \langle a, b | a^2 = b^2 = e, a b = b a \rangle. \end{equation*}

As a more interesting example, let us look at the so-called modular group, which is important in various areas of physics, such as string theory and conformal field theory. Consider \(2 \times 2\) matrices \(M = \begin{pmatrix} a \amp b \\ c \amp d \end{pmatrix}\) with integer entries and such that \(\det M = 1\text{.}\) This group is denoted by \(SL(2, \mathbb{Z})\text{,}\) the special linear group of \(2 \times 2\) matrices with integer entries. If we also impose the identification \(M = - M\) in \(SL(2, \mathbb{Z})\) we end up with the group \(PSL(2,\mathbb{Z})\text{,}\) the projective special linear group of \(2 \times 2\) matrices with integer entries, which is also known as the modular group. This group is important because it is the group of linear fractional transformations of the upper half of the complex plane, where it acts as:

\begin{equation} z \mapsto \frac{a z + b}{c z + d},\label{equation-modular}\tag{1.5.1} \end{equation}

for \(z \in \mathbb{C}\) with \(\Im(z) \gt 0\text{.}\)

This group looks rather complicated, but in fact one can show that all such \(2 \times 2\) matrices can be obtained by repeatedly multiplying together two matrices:

\begin{equation*} S = \begin{pmatrix} 0 \amp 1 \\ -1 \amp 0 \end{pmatrix}, \qquad T = \begin{pmatrix} 1 \amp 1 \\ 0 \amp 1 \end{pmatrix}. \end{equation*}

In terms of fractional linear transformations, what that means is that the general transformation (1.5.1) can be obtained by repeatedly composing the two transformations:

\begin{equation*} S: z \mapsto - \frac{1}{z}, \qquad T: z \mapsto z + 1. \end{equation*}

Moreover, it is clear that \(S^2 = I\text{,}\) where \(I\) is the identity matrix (the identity element in the group), and one can check easily that \((S T)^3 = I\) as well. Thus we can write the presentation of \(PSL(2,\mathbb{Z})\) as:

\begin{equation*} \langle S, T | S^2 = I, (S T)^3 = I \rangle. \end{equation*}
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