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Chapter 6: Problem 3

Let \(p\) be a prime number. The group \(\mathrm{GL}(2, \mathbb{Z} / p \mathbb{Z})\) has \(\left(p^{2}-1\right)\left(p^{2}-p\right)\) elements. Hint. How many first columns are possible. How often can a given column be extended to an invertible matrix? Deduce from this that the group \(\operatorname{SL}(2, \mathbb{Z} / p \mathbb{Z})\) has \(\left(p^{2}-1\right) p\) elements.

Short Answer

Expert verified
The group \(\operatorname{SL}(2, \mathbb{Z} / p \mathbb{Z})\) has \((p^{2}-1) p\) elements.

Step by step solution

01

Understanding GL(2, \(\mathbb{Z} / p \mathbb{Z}\))

\(\mathrm{GL}(2, \mathbb{Z} / p \mathbb{Z})\) is the group of 2x2 invertible matrices over the field \(\mathbb{Z} / p \mathbb{Z}\). For a matrix to be invertible, its determinant must be non-zero in this field.
02

Counting the First Column Options

The first column of a matrix in \(\mathrm{GL}(2, \mathbb{Z} / p \mathbb{Z})\) can be any of the \(p^2\) column vectors except the zero vector. This gives \(p^{2} - 1\) choices for the first column.
03

Extending to an Invertible Matrix

For any chosen first column, the second column must make the matrix invertible. The determinant of \(\begin{pmatrix} a & b \ c & d \end{pmatrix}\) is \(ad - bc\). Given the first column \(\begin{pmatrix} a \ c \end{pmatrix}\), we require that not all \(b\) and \(d\) satisfy \(ad - bc = 0\), meaning any choice for the second column as long as it is linearly independent to the first gives \(p^{2} - p\) valid options.
04

Total Elements in \(GL(2, \mathbb{Z} / p \mathbb{Z})\)

By combining these choices, the total number of elements in \(\mathrm{GL}(2, \mathbb{Z} / p \mathbb{Z})\) is the product of the options for the first column and the second column: \((p^{2} - 1)(p^{2} - p)\).
05

Counting Elements in SL(2, \(\mathbb{Z} / p \mathbb{Z}\))

\(\operatorname{SL}(2, \mathbb{Z} / p \mathbb{Z})\) is the subgroup of matrices from \(\mathrm{GL}(2, \mathbb{Z} / p \mathbb{Z})\) with determinant 1. Given the total number of invertible matrices, \(\operatorname{SL}(2, \mathbb{Z} / p \mathbb{Z})\) has \((p^{2} - 1)\cdot p\) elements because exactly \(1/p\) of matrices in \(\mathrm{GL}(2, \mathbb{Z} / p \mathbb{Z})\) will have a determinant of 1.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

GL(2, Z/pZ)
The group \(\mathrm{GL}(2, \mathbb{Z} / p \mathbb{Z})\) consists of all 2x2 invertible matrices that can be formed over the finite field \(\mathbb{Z} / p \mathbb{Z}\). Here, \(p\) is a prime number, which means we have a set of numbers \(\{0, 1, 2, ..., p-1\}\) obeying arithmetic modulo \(p\).
To understand these matrices, remember two things:
  • The matrices are 2x2, meaning they look like: \(\begin{pmatrix} a & b \ c & d \end{pmatrix}\).
  • A matrix is invertible if there exists another matrix which, when multiplied, results in the identity matrix.
This set forms a group under matrix multiplication, notably because the determinant dictates if a matrix is invertible. We use these matrices to explore linear transformations over finite fields, providing a structured "playground" for mathematicians to examine various algebraic properties.
SL(2, Z/pZ)
The special linear group \(\operatorname{SL}(2, \mathbb{Z} / p \mathbb{Z})\) is a subset of \(\mathrm{GL}(2, \mathbb{Z} / p \mathbb{Z})\) consisting of matrices with a determinant of 1. The determinant here is a crucial concept, helping to maintain the structure and the properties of linear transformations.
  • Why determinant 1? This condition ensures the matrix scales volume in space by exactly 1, implementing transformations that are entirely area-preserving.
  • Finding the number of such matrices involves calculating that a fraction (specifically \(\frac{1}{p}\)) of matrices in \(\mathrm{GL}(2, \mathbb{Z} / p \mathbb{Z})\) qualifies.
Therefore, the total elements in \(\operatorname{SL}(2, \mathbb{Z} / p \mathbb{Z})\) equals the terms \((p^{2}-1) \cdot p\), which aligns with preserving determinant properties during multiplication.
Invertible Matrices
An invertible matrix is one that can "undo" its own transformation, meaning when a matrix \(A\) is multiplied by its inverse \(A^{-1}\), we get the identity matrix \(I\).
The definition of being invertible is tied directly to the determinant: if and only if the determinant is non-zero (in the field \(\mathbb{Z} / p \mathbb{Z}\)), the matrix is invertible.
This determinant condition is crucial in building the sets \(\mathrm{GL}(2, \mathbb{Z} / p \mathbb{Z})\) and \(\operatorname{SL}(2, \mathbb{Z} / p \mathbb{Z})\), as it filters out matrices that effectively cannot be reversed, a vital interest in transformations and solving linear equations.
Determinant
The determinant of a 2x2 matrix \(\begin{pmatrix} a & b \ c & d \end{pmatrix}\) is calculated as \(ad - bc\). It gives us valuable information about matrix properties, such as invertibility.
  • If the determinant is non-zero, the matrix is invertible, which is paramount for the groups \(\mathrm{GL}(2, \mathbb{Z} / p \mathbb{Z})\) and \(\operatorname{SL}(2, \mathbb{Z} / p \mathbb{Z})\).
  • For matrices in \(\operatorname{SL}(2, \mathbb{Z} / p \mathbb{Z})\), the determinant must specifically be 1, making them special cases of invertible matrices.
Understanding the determinant helps one visualize the scaling transformations that these matrices represent, particularly how they impact areas and volumes in linear algebraic operations.
Finite Fields
Finite fields, notably \(\mathbb{Z} / p \mathbb{Z}\), are arithmetic structures consisting of a finite number of elements governed by operations of addition and multiplication modulo a prime \(p\).
These fields are fundamental in understanding \(\mathrm{GL}(2, \mathbb{Z} / p \mathbb{Z})\) and \(\operatorname{SL}(2, \mathbb{Z} / p \mathbb{Z})\) because they define the entries of matrices and dictate how operations like the determinant work.
  • Working within \(\mathbb{Z} / p \mathbb{Z}\), numbers wrap around after reaching \(p\), creating periodic behavior ideal for cryptography, coding theory, and error-checking algorithms.
  • Finite fields provide a simplified context for exploring complex algebraic concepts, crucial in number theory and algebraic structures.
Through finite fields, matrices inherit concrete rules, offering a rich field of study for combinatorial and algebraic research.

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Most popular questions from this chapter

Let \(G \subset \operatorname{SL}(2, \mathbb{R})\) be a finite subgroup, such that its elements admit a common fixed point in H. (One can show that any finite, or more general any compact, subgroup \(G \subset \cdot S L(2, \mathbb{R})\) has this property!) Show that \(G\) is cyclic.

Let \(q_{1}\) and \(q_{2}\) be two relatively prime natural numbers. The Chinese Remainder Theorem claims that the natural homomorphism \(\mathbb{Z} / q_{1} q_{2} \mathbb{Z} \rightarrow \mathbb{Z} / q_{1} \mathbb{Z} \times \mathbb{Z} / q_{2} \mathbb{Z}\) is an isomorphism. Deduce from this that the natural homomorphism $$ \mathrm{GL}\left(2, \mathbb{Z} / q_{1} q_{2} \mathbb{Z}\right) \longrightarrow \mathrm{GL}\left(2, \mathbb{Z} / q_{1} \mathbb{Z}\right) \times \mathrm{GL}\left(2, \mathbb{Z} / q_{2} \mathbb{Z}\right) $$ is an isomorphism.

The derivative of a modular function is a meromorphic modular form of weight \(2 .\)

Let $$ f(z)=\sum_{n=1}^{\infty} a_{n} e^{2 \pi i n z} $$ be a cusp form of weight \(k\). Prove an estimate of the kind $$ \left|a_{n}\right| \leq C n^{k / 2} $$ (E. HECKE, 1927\()\) with a suitable constant \(C .\) Hint. Use the integral representation for the FOUR1ER coefficients, and apply the estimate $$ |f(z)| \leq C^{\prime} y^{-k / 2} $$ for the special value \(y=1 / n\) P. DELIGNE proved in 1974 the RAMANUJAN-PETERSSON conjecture, which ensures the much stronger estimate $$ \left|a_{n}\right| \leq C(\varepsilon) n^{(k-1) / 2+\varepsilon} \text { for any } \varepsilon>0 $$

For any point \(a \in \mathbb{H}\) there exists an entire modular form (even of weight 12 ), which vanishes at \(a\), but does not vanish identically. Hint. Use the knowledge of the zeros of \(\Delta\).
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