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Magazine Chapter 6: Problem 2
The natural group homomorphism $$ \mathrm{SL}(2, \mathbb{Z}) \longrightarrow \mathrm{SL}(2, \mathbb{Z} / q \mathbb{Z}) $$ is surjective. In particular, $$ [\Gamma: \Gamma[q]]=\\# \operatorname{SL}(2, \mathbb{Z} / q \mathbb{Z}) $$
Short Answer
Expert verified
The homomorphism is surjective and \([\Gamma: \Gamma[q]] = \# \operatorname{SL}(2, \mathbb{Z}/q\mathbb{Z})\).
Step by step solution
01
Understanding the Problem
We need to show that the natural homomorphism from \( \mathrm{SL}(2, \mathbb{Z}) \) to \( \mathrm{SL}(2, \mathbb{Z}/q\mathbb{Z}) \) is surjective. This means every element in \( \mathrm{SL}(2, \mathbb{Z}/q\mathbb{Z}) \) has a preimage in \( \mathrm{SL}(2, \mathbb{Z}) \). Additionally, we need to verify the index equivalence \( [\Gamma: \Gamma[q]] = \# \operatorname{SL}(2, \mathbb{Z}/q\mathbb{Z}) \), where \( \Gamma = \mathrm{SL}(2, \mathbb{Z}) \) and \( \Gamma[q] \) is the kernel of this homomorphism.
02
Set Up the Homomorphism
Consider the homomorphism \( \phi: \mathrm{SL}(2, \mathbb{Z}) \rightarrow \mathrm{SL}(2, \mathbb{Z}/q\mathbb{Z}) \) defined by reducing the entries of a matrix modulo \( q \). For a matrix \( A \in \mathrm{SL}(2, \mathbb{Z}) \), \( \phi(A) \in \mathrm{SL}(2, \mathbb{Z}/q\mathbb{Z}) \) because it retains the property that \( \det(A) = 1 \), now considered in \( \mathbb{Z}/q\mathbb{Z} \).
03
Show Surjectivity
We need to demonstrate that every matrix in \( \mathrm{SL}(2, \mathbb{Z}/q\mathbb{Z}) \) is an image under \( \phi \) of some matrix in \( \mathrm{SL}(2, \mathbb{Z}) \). This proof involves showing that for any matrix \( B \in \mathrm{SL}(2, \mathbb{Z}/q\mathbb{Z}) \), we can find a matrix in \( \mathrm{SL}(2, \mathbb{Z}) \) which reduces to \( B \). This is achievable as according to standard results in arithmetic group theory, the reduction map is always surjective for modular arithmetic over integers.
04
Verify the Index
The index \( [\Gamma: \Gamma[q]] \) represents the number of distinct cosets of \( \Gamma[q] \) in \( \Gamma \). Since \( \phi \) is surjective, \( \Gamma / \Gamma[q] \cong \operatorname{SL}(2, \mathbb{Z}/q\mathbb{Z}) \) as groups. Thus, the index is equal to the order of \( \operatorname{SL}(2, \mathbb{Z}/q\mathbb{Z}) \), proving that \( [\Gamma: \Gamma[q]] = \# \operatorname{SL}(2, \mathbb{Z}/q\mathbb{Z}) \).
05
Conclusion
Hence, we have shown that the natural group homomorphism \( \mathrm{SL}(2, \mathbb{Z}) \longrightarrow \mathrm{SL}(2, \mathbb{Z}/q\mathbb{Z}) \) is surjective. Consequently, the index \( [\Gamma: \Gamma[q]] \) is equal to the cardinality of \( \mathrm{SL}(2, \mathbb{Z}/q\mathbb{Z}) \), confirming the given exercise statement.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Modular Arithmetic
Modular arithmetic is like clock arithmetic. It's a system where numbers wrap around after reaching a certain value, called the modulus. For instance, imagine a clock where after 12, it resets to 1. Similarly, in modular arithmetic, numbers wrap around after a specified modulus.
- This type of arithmetic is essential for computations involving a 'moduli', which could be any positive integer.
- When we say a number is equivalent "modulo some number", we mean after division by that number (the modulus), the remainder is the same.
- For example, in modulo 5 arithmetic: 7, 12, and 17 are equivalent because they leave a remainder of 2 when divided by 5.
SL(2, Z)
The term \( \mathrm{SL}(2, \mathbb{Z}) \) represents the special linear group of 2x2 integer matrices with determinant 1.
- This group plays a crucial role in various areas of mathematics like geometry and number theory.
- Matrices in \( \mathrm{SL}(2, \mathbb{Z}) \) transform a two-dimensional integer grid but preserve area because their determinant is 1.
- To be in \( \mathrm{SL}(2, \mathbb{Z}) \), a matrix \( A = \begin{pmatrix} a & b \ c & d \end{pmatrix} \) must satisfy \( ad - bc = 1 \).
Surjective Mapping
A map or function is deemed surjective (or onto) if every element in the target set is mapped by at least one element from the domain.
- Surjectivity assures that the mapping is covering the entire target set, leaving no element unmapped.
- In our context, it indicates that every matrix in \( \mathrm{SL}(2, \mathbb{Z}/q\mathbb{Z}) \) comes from some matrix in \( \mathrm{SL}(2, \mathbb{Z}) \).
- This property is pivotal when proving that two groups are essentially structured in a certain way under a given mapping, revealing relationships between their elements.
Kernel of a Homomorphism
The kernel of a homomorphism is a subset of elements in the domain that map to the identity element in the codomain.
- It's a measure of how far the homomorphism is from being an isomorphism.
- For a homomorphism \( \phi: G \to H \), the kernel is defined as \( \text{ker}(\phi) = \{ g \in G \mid \phi(g) = e_H \} \), where \( e_H \) is the identity element in \( H \).
- The kernel's size or structure often provides information about the nature of the surjective mapping, helping identify whether or not the mapping is injective.
