Problem 22 - Algebra

Question

Let \(p\) be a prime \(\geqq 5\). Let \(G\) be a subgroup of \(S L_{2}\left(\mathbf{Z} / p^{n} \mathbf{Z}\right)\) with \(n \geq 1\). Assume that the image of \(G\) in \(S L_{2}(\mathbf{Z} / p \mathbf{Z})\) under the natural homomorphism is all of \(S L_{2}(\mathbf{Z} / p \mathbf{Z})\). Prove that \(G=S L_{2}\left(\mathbf{Z} / p^{n} Z\right)\).

Step-by-Step Solution

Answer

To prove that \(G = SL_{2}\left(\mathbf{Z} / p^{n} \mathbf{Z}\right)\), we show that \(G\) contains all the elementary matrices of \(SL_{2}\left(\mathbf{Z} / p^{n} \mathbf{Z}\right)\), i.e., \(E_1(p^n)\) and \(E_2(p^n)\), which is sufficient to generate the entire group. By defining appropriate subgroups and using matrix multiplication, we demonstrate that \(G\) indeed contains these elementary matrices, thus concluding that \(G\) equals \(SL_{2}\left(\mathbf{Z} / p^{n} \mathbf{Z}\right)\).

Step 1: Define elementary matrices

For \(SL_{2}\left(\mathbf{Z} / p^{n} \mathbf{Z}\right)\), we will focus on the elementary matrices, defined as follows: 1. \(E_{1}(k) = \begin{pmatrix} 1 & k \\ 0 & 1 \end{pmatrix} \) 2. \(E_{2}(k) = \begin{pmatrix} 1 & 0 \\ k & 1 \end{pmatrix} \) where \(k \in \mathbf{Z} / p^{n} \mathbf{Z}\).

Step 2: Lift to \(GL_2(\mathbf{Z/p^{n+1}Z})\)

Let's define the following subgroups: \(H_1 := \{g\in SL_2(\mathbf{Z}/p\mathbf{Z}): g \equiv E_2(1) \mod p\}\) and \(H_2 := \{g\in SL_2(\mathbf{Z}/p^{n+1}\mathbf{Z}): g\equiv E_2(p^n) \mod p\}\). As \(H_1 \subseteq G\), and the natural homomorphism is surjective, we have \(H_1\cap H_2\) is non-trivial. Let \(g\) be a non-trivial element in \(H_1 \cap H_2\). By definition, \(g\equiv E_2(p^n) \mod p^{n+1}\) and \(g\equiv E_2(1) \mod p\), thus \(g= E_2(1+p^{n}t)\) for some \(t\in \mathbf{Z}/p^n\mathbf{Z}\).

Step 3: Prove \(G\) contains \(E_1(p^n)\)

Now, consider the matrix product: \(g E_1(-t)g^{-1} E_1(t) = E_2(1+p^n t)E_1(-t)E_2(-1+p^n(-1-t))E_1(t)\) Using matrix multiplication, we get \(g E_1(-t)g^{-1} E_1(t) = E_1(p^n)\). Since \(g\) and \(E_1(t)\) are in \(G\) (and hence their product and inverse are also in \(G\)), we have \(E_1(p^n) \in G\).

Step 4: Prove \(G\) contains \(E_2(p^n)\)

We have shown that \(E_2(1+p^n t)\) is in \(G\) for all \(t\in \mathbf{Z}/p^n\mathbf{Z}\). The product of these matrices is as follows: \(E_2(1+p^n t)E_1(-p^{n-1}t) = E_2(1 + p^n(t - p^{n-1}t)) = E_2(p^n)\). We have shown that \(E_2(p^n) \in G\).

Step 5: Conclude \(G\) equals \(SL_{2}\left(\mathbf{Z} / p^{n} \mathbf{Z}\right)\)

We have shown that \(G\) contains \(E_1(p^n)\) and \(E_2(p^n)\). It is well-known that a group containing all elementary matrices of type \(E_1\) and \(E_2\) generates the entire group of \(SL_{2}\left(\mathbf{Z} / p^{n} \mathbf{Z}\right)\). Therefore, we can now conclude that \(G = SL_{2}\left(\mathbf{Z} / p^{n} \mathbf{Z}\right)\).

Key Concepts

Group Theory

The study of groups, which are sets equipped with an operation that combines any two elements to form a third element while adhering to four fundamental principles: closure, associativity, the presence of an identity element, and the presence of inverse elements.

In the context of the exercise, we delve into a specific group known as the special linear group, denoted by SL2, which consists of 2x2 matrices with determinant equal to one. The group operation in this case is matrix multiplication. The interesting aspect of this group is the way it changes when we consider it over different types of number systems, such as the integers modulo a prime power, indicated by \(Z/p^nZ\).

Understanding the structure of groups like SL2 over modular arithmetic systems is important in several areas of mathematics including number theory, cryptography, and geometry. The task involves demonstrating that a certain subgroup 'G' is actually the whole group SL2 under a specific modular context, using properties of elementary matrices and group theory concepts.

Elementary Matrices

Elementary matrices are building blocks for matrix operations particularly useful in group theory and linear algebra. These matrices represent simple row operations that can be applied to any matrix. The operations include adding a multiple of one row to another, swapping two rows, or multiplying a row by a non-zero element.

In the special linear group SL2, which we focus on in the exercise, elementary matrices take a simpler form since they only involve adding multiples of one row to another due to the determinant being restricted to 1. This special case involves matrices of the type \(E_1(k)\) and \(E_2(k)\) as defined in the solution steps.

These matrices are important because they help prove that the subgroup 'G' is as large as it can be, by showing that it contains these types of matrices for particular values of 'k'. Hence, illustrating that 'G' contains all the necessary elements to generate the entire group SL2 modulo \(p^n\).

Modular Arithmetic

Also known as clock arithmetic, modular arithmetic is a system of arithmetic for integers, where numbers wrap around after reaching a certain value—the modulus. It's like looking at the remainder when one number is divided by another (the modulus).

When we talk about SL2 over \(Z/p^nZ\), we are considering matrices whose elements are not just integers but are integers under this modular arithmetic system. This has profound implications on the structure and properties of the group, and especially on the concept of elementary matrices within this group.

The concept is pivotal in understanding the operations within the group 'G' and how it relates to the whole group SL2. It is particularly interesting to see how operations extend naturally to group elements when considering them modulo \(p\) and \(p^n\), and how this plays a key role in proving the subgroup in question is indeed the entire group. The clever use of modular arithmetic in the solution steps is central to bridging the gap between the subgroup 'G' and the entirety of SL2 within the modular context.