Problem 20 - Algebra

Question

Show that one has an exact sequence $$ 1 \rightarrow S L_{2}(\mathbf{Z} / N \mathbf{Z}) \rightarrow G L_{2}(\mathbf{Z} / N \mathbf{Z}) \stackrel{d e !}{\leftrightarrow}(\mathbf{Z} / N \mathbf{Z})^{*} \rightarrow 1 . $$ In fact, show that $$ G L_{2}(\mathbf{Z} / \mathrm{NZ})=S L_{2}(\mathbf{Z} / N Z) G_{N} $$ where \(G_{N}\) is the group of matrices $$ \left(\begin{array}{ll} 1 & 0 \\ 0 & d \end{array}\right) \text { with } d \in(\mathbf{Z} / N Z)^{\bullet} $$

Step-by-Step Solution

Answer

We show the existence of an exact sequence by proving that the determinant is a homomorphism from \(GL_2(\mathbf{Z}/N\mathbf{Z})\) to \((\mathbf{Z}/N\mathbf{Z})^*\) and that its kernel is \(SL_2(\mathbf{Z}/N\mathbf{Z})\). After we demonstrate that any element in \(GL_2(\mathbf{Z}/N\mathbf{Z})\) can be expressed as a product of an element in \(SL_2(\mathbf{Z}/N\mathbf{Z})\) and an element of \(G_N\), proving that \(GL_2(\mathbf{Z}/N\mathbf{Z})=SL_2(\mathbf{Z}/N\mathbf{Z})G_N\). The exact sequence is thus: \( 1 \rightarrow SL_2(\mathbf{Z}/N\mathbf{Z}) \rightarrow GL_2(\mathbf{Z}/N\mathbf{Z}) \stackrel{d\mathrm{e!}}{\rightarrow} (\mathbf{Z}/N\mathbf{Z})^* \rightarrow 1 \).

Step 1: Show that determinant is a homomorphism

Let's first show that \(d\mathrm{e!}\) is a homomorphism from \(GL_2(\mathbf{Z}/N\mathbf{Z})\) to \((\mathbf{Z}/N\mathbf{Z})^*\). Since the determinant is multiplicative (i.e. \(\det(AB) = \det(A)\det(B)\)), and \(GL_2(\mathbf{Z}/N\mathbf{Z})\) consists of matrices with nonzero determinant, the image of the determinant function will be in \((\mathbf{Z}/N\mathbf{Z})^*\), as required.

Step 2: Determine the kernel of the determinant homomorphism

Now, let's find the kernel of the determinant homomorphism. The kernel is the set of matrices in \(GL_2(\mathbf{Z}/N\mathbf{Z})\) with determinant equal to 1. These are precisely the matrices in \(SL_2(\mathbf{Z}/N\mathbf{Z})\). Thus, we have the kernel of the determinant homomorphism as \(SL_2(\mathbf{Z}/N\mathbf{Z})\). This allows us to create the exact sequence as follows: $$ 1 \rightarrow SL_2(\mathbf{Z}/N\mathbf{Z}) \rightarrow GL_2(\mathbf{Z}/N\mathbf{Z}) \stackrel{d\mathrm{e!}}{\rightarrow} (\mathbf{Z}/N\mathbf{Z})^* \rightarrow 1 . $$

Step 3: Show that \(GL_2(\mathbf{Z}/N\mathbf{Z})=SL_2(\mathbf{Z}/N\mathbf{Z})G_N\)

To show that \(GL_2(\mathbf{Z}/N\mathbf{Z})=SL_2(\mathbf{Z}/N\mathbf{Z})G_N\), let \(A\in GL_2(\mathbf{Z}/N\mathbf{Z})\) be any given matrix. We can write \(A\) as $$ A = \left(\begin{array}{cc} a & b \\ c & d \end{array}\right),$$ with \(ad - bc \neq 0\) mod \(N\). Let's consider the matrix $$ B = \left(\begin{array}{cc} a & b \\ c & d+\frac{ad-bc}{\det{A}} \end{array}\right) = \left(\begin{array}{cc} a & b \\ c & d' \end{array}\right)$$ where \(d' \equiv d+\frac{ad-bc}{\det{A}}\) mod \(N\). Notice that \(B\) has a determinant of 1, so \(B\in SL_2(\mathbf{Z}/N\mathbf{Z})\). Now consider the matrix $$ C = \left(\begin{array}{cc}1 & 0 \\ 0 & \det{A}\end{array}\right) \in G_N. $$ We can see that \(A = BC\), since: $$ BC = \left(\begin{array}{cc} a & b \\ c & d' \end{array}\right)\left(\begin{array}{cc}1 & 0 \\ 0 & \det{A}\end{array}\right) = \left(\begin{array}{cc} a & b \\ c & d \end{array}\right) = A. $$ This shows that any given element of \(GL_2(\mathbf{Z}/N\mathbf{Z})\) can be expressed as a product of an element in \(SL_2(\mathbf{Z}/N\mathbf{Z})\) and an element of \(G_N\), which proves the required result: $$GL_2(\mathbf{Z}/N\mathbf{Z})=SL_2(\mathbf{Z}/N\mathbf{Z})G_N.$$

Key Concepts

SL2 and GL2 groups

Understanding SL2 and GL2 groups is crucial for diving into higher algebra concepts. Let's start with GL2, the General Linear Group of degree 2, defined over a field, or in this case, the integers modulo N, denoted as \(GL_2(\mathbf{Z}/N\mathbf{Z})\). This group consists of all 2x2 invertible matrices, with elements taken from \(\mathbf{Z}/N\mathbf{Z}\), and the group operation is matrix multiplication.

On the other hand, the SL2, or Special Linear Group of degree 2, is a subgroup of GL2, represented as \(SL_2(\mathbf{Z}/N\mathbf{Z})\). It includes all matrices from the GL2 group but with a specific condition that their determinants must be 1. This property is fundamentally linked to how these matrices preserve area and volume in geometrical transformations.

When examining the exact sequence involving these groups, we gain insight into how these structures relate to each other through the properties of their determinants. The sequence effectively splits GL2 into components detailing the inherent symmetries and the scaling transformations represented by SL2 and the matrices which adjust determinant values within \(GL2\).

Determinant Homomorphism

A determinant homomorphism is at the heart of our exact sequence problem. In mathematics, a homomorphism is a structure-preserving map between two algebraic structures, such as groups or rings. The determinant function, denoted \(de!\), acts as a homomorphism when it maps matrices from \(GL_2(\mathbf{Z}/N\mathbf{Z})\) to the multiplicative group of units \( (\mathbf{Z} / N \mathbf{Z})^{*}\), which contains the elements that have a multiplicative inverse modulo N.

The key properties of this homomorphism are its ability to transform the matrix multiplication into multiplication of their determinants and to pinpoint matrices of a particular form—those with determinant 1—as constituting the kernel of this homomorphism. This kernel, \(SL_2(\mathbf{Z}/N\mathbf{Z})\), provides a clear pivot point around which the exact sequence is structured, highlighting the connection between the structures within the sequence.

Subsequently, understanding determinant homomorphism helps us see how the characteristics of \(GL_2\) are decomposed into simpler components.

Group Theory in Mathematics

An overarching field that encompasses these concepts is group theory in mathematics. This area of abstract algebra deals with the study of algebraic structures known as groups, which provide a unifying framework for analyzing symmetrical structures in varying mathematical contexts.

Groups consist of a set equipped with an operation that combines any two of its elements to form another element, satisfying four fundamental properties: closure, associativity, identity, and invertibility. To show the synergy between the concepts, group theory enables us to formalize and study the properties of SL2 and GL2 groups and their interrelationships.

The exact sequence from our exercise exhibits deep interconnections between these groups, illuminated by group theory. The sequence communicates not just a collection of individual entities but also the transformation between them, characterized by certainty and predictability that comes through both homomorphism and the structural integrity of groups. This intersection of algebraic objects offers a window into the symmetries and invariants that form the basis for modern algebraic understanding.