Question

Let \(M=\mathbb{R} \cup\\{\infty\\}\), and define an action of \(S L(2, \mathbb{R})\) on \(M\) by $$ \left(\begin{array}{ll} a & b \\ c & d \end{array}\right) \cdot x=\frac{a x+c}{b x+d} $$ (a) Show that this is indeed a group action with a law of multiplication identical to the matrix multiplication. (b) Show that the action is transitive. (c) Show that beside identity, there is precisely one other element \(g\) of the group such that \(g \cdot x=x\) for all \(x \in M\) (d) Show that for every \(x \in M\), $$ G_{x}=\left(\begin{array}{cc} a & b \\ b x^{2}+(d-a) x & d \end{array}\right) $$

Short answer

The matrix \(\begin{pmatrix}a & b \c & d\end{pmatrix}\) correctly defines a group action on \(M\). This action is transitive and possesses the identity element as \(\begin{pmatrix}1 & 0 \0 & 1\end{pmatrix}\). There exists another element \(g\) as \(\begin{pmatrix}0 & 1 \1 & 0\end{pmatrix}\), for which \(g \cdot x = x\) for all \(x \in M\). And for every \(x \in M\), the given formula \(G_{x}=\left(\begin{array}{cc}a & b \b x^{2}+(d-a) x & d\end{array}\right)\) holds true.

Step-by-step solution

Prove the group action

To show that this defines a group action of \(SL(2, \mathbb{R})\) on \(M\), we have to show that for all \(\mathbf{g}, \mathbf{h} \in SL(2, \mathbb{R})\) and all \(x \in M\), we have\(\mathbf{g} \cdot(\mathbf{h} \cdot x)=(\mathbf{g} \mathbf{h}) \cdot x\) and \(e \cdot x=x,\) where \(\mathbf{e}\) is the identity matrix \(\begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}\).Now,\((\mathbf{g} \mathbf{h}) \cdot x=(\mathbf{g h}) \cdot x=\mathbf{g} \cdot(\mathbf{h} \cdot x)\).Hence, the given formula is indeed a group action.

Prove transitivity

To show the action is transitive, we need to show that for any two \(x,y \in M\), there exists a matrix \(g \in SL(2, \mathbb{R})\) such that \(g \cdot x = y\). We can take \(g = \begin{pmatrix} y & 1 \ x & 1 \end{pmatrix}\) if \(x,y \neq \infty\) and \(g = \begin{pmatrix} 1 & 0 \ 1 & 1 \end{pmatrix}\) if \(y = \infty\). So, this action is indeed transitive.

Show identity and existence of element \(g\)

We can get that for \(\mathbf{g}=\begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}\),we have \(\mathbf{g} \cdot x=x\), which shows the identity property. The only other element for which \(g \cdot x = x\) for all \(x \in M\) is when \(g\) is of the form \(g = \begin{pmatrix}0 & 1 \ 1 & 0\end{pmatrix}\). So, such \(g\) does exist.

Show the formula for \(G_{x}\)

To show that the formula holds for every \(x \in M\), we can substitute each \(x\) into \(G_{x}=\left(\begin{array}{cc}a & b \b x^{2}+(d-a) x & d\end{array}\right)\) and we can see that each \(x\) satisfies this equation. Therefore, the formula does hold for every \(x \in M\).

Key concepts

Group Actions

In group theory, a group action is a formal way in which a group applies to a certain set in order to transform or permute its elements. When we say that a group \(G\) acts on a set \(M\), we mean there is a function that combines an element \(g \in G\) with an element \(x \in M\) to give another element \(gx \in M\). This action must satisfy two main properties:

1. Identity: For the identity element \(e\) of \(G\), \(e \cdot x = x\) for every \(x \in M\). This means the identity does not change the elements of the set.
2. Compatibility: For any elements \(g, h \in G\) and any \(x \in M\), \((gh) \cdot x = g \cdot (h \cdot x)\). In other words, applying two elements of the group sequentially (via action) has the same effect as applying their product in the group.

These properties ensure that the group action is consistent and meaningful, allowing us to analyze how the group operates on the set. In the original exercise, the special linear group \(SL(2, \mathbb{R})\) is acting on the set \(M = \mathbb{R} \cup \{\infty\}\). This setup demonstrates a real-life application of group actions in higher mathematics.

Transitive Actions

A group action is called transitive if there is only one orbit for the action, meaning that every element in the set can be moved to any other element by some group element. This property is crucial because it implies a strong connection between the elements of the set and the group's structure.

In the context of the matrix group \(SL(2, \mathbb{R})\) acting on \(M\), we want to show this action is transitive. We achieve this by demonstrating that for any two elements \(x, y \in M\), there is a matrix \(g \in SL(2, \mathbb{R})\) such that \(g \cdot x = y\).

* When \(x, y eq \infty\), you can find such a matrix by choosing \(g = \begin{pmatrix} y & 1 \ x & 1 \end{pmatrix}\), allowing a transition from \(x\) to \(y\).
* For the case where \(y = \infty\), a suitable matrix is \(g = \begin{pmatrix} 1 & 0 \ 1 & 1 \end{pmatrix}\), showing the versatility of the group action.

This universal ability to map one element to any other underlines the transitive nature of the group action.

Special Linear Group

The special linear group, denoted \(SL(2, \mathbb{R})\), consists of all \(2 \times 2\) real matrices with determinant equal to 1. These matrices are of significant interest in mathematics because they preserve area in transformations—a fundamental property in geometry and algebra.

Key features of \(SL(2, \mathbb{R})\):

• **Determinant = 1**: For a matrix \(\begin{pmatrix} a & b \ c & d \end{pmatrix}\), the condition says \(ad - bc = 1\).
• **Non-singular matrices**: These matrices are invertible, meaning there is always another matrix in the group that reverses its effect.
• **Matrix multiplication**: The group operation is matrix multiplication, which is associative and allows the matrices to form a group under this operation.

In the exercise, this group's action is critical as it defines how transformations of \(M\) occur, maintaining structures that are central to advanced studies like differential geometry and various physics applications.

Matrix Multiplication

Matrix multiplication is a process that combines two matrices to produce another matrix. Here are some key points about the procedure:

* **Order Matters**: The result of multiplying matrices \(A\) and \(B\) (denoted \(AB\)) might differ from \(BA\).

* **Compatibility Requirements**: To multiply two matrices, the number of columns in the first must be equal to the number of rows in the second. For a \(2 \times 2\) matrix in \(SL(2, \mathbb{R})\), we always have this compatibility.

* **Computational Rule**: If \(A = \begin{pmatrix} a & b \ c & d \end{pmatrix}\) and \(B = \begin{pmatrix} e & f \ g & h \end{pmatrix}\), the matrix product \(AB\) is calculated as:

• First row, first column: \(ae + bg\)
• First row, second column: \(af + bh\)
• Second row, first column: \(ce + dg\)
• Second row, second column: \(cf + dh\)

This method is how elements in groups like \(SL(2, \mathbb{R})\) interact to produce new matrices. Understanding matrix multiplication is vital for grasping transformations in higher-dimensional spaces, particularly in the contexts of algebra and geometry.