Problem 21 - Algebra: Pure and Applied

Question

What is the order of \(G L\left(2, \mathbb{Z}_{2}\right) ?\)

Step-by-Step Solution

Answer

The order of \(GL(2, \mathbb{Z}_2)\) is 6.

Step 1: Define the General Linear Group

The general linear group, \(GL(2, \mathbb{Z}_2)\), consists of all \(2 \times 2\) invertible matrices with entries from the field \(\mathbb{Z}_2\). \(\mathbb{Z}_2\) represents the integers modulo 2, which are \{0, 1\}.

Step 2: Determine Matrix Entries

Since the matrices are \(2 \times 2\), and each entry can be either 0 or 1 (the elements of \(\mathbb{Z}_2\)), there are \(2^4 = 16\) possible matrices.

Step 3: Calculate the Invertibility Condition

A matrix is invertible if its determinant is non-zero. For a matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \), the determinant is \(ad - bc\) modulo 2. The matrix is invertible if \(ad - bc \equiv 1 \pmod{2}\).

Step 4: Count Invertible Matrices

We must count the number of matrices for which \(ad - bc \equiv 1\). If \(a = 0\), then \(c\) cannot be 0, and so forth. For different configurations of 0 and 1 from each matrix position, calculate which satisfy the invertibility condition.

Step 5: Verify the Calculation

Through enumerating possibilities or using formula-based counting, verify that the number of invertible \(2 \times 2\) matrices is 6. This confirms past calculations and common knowledge about \(GL(2, \mathbb{Z}_2)\).

Key Concepts

Invertible Matrices

An invertible matrix is a square matrix that possesses an inverse. More simply put, it is a matrix that can be "undone" to obtain the identity matrix when multiplied with its inverse.
To check if a matrix is invertible, we mainly need to consider its determinant:

• If the determinant is non-zero, the matrix is invertible.
• If the determinant equals zero, the matrix is not invertible.

For example, a matrix of the form \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \) is invertible if \( ad - bc eq 0 \). In the context of modulo arithmetic like in \( \mathbb{Z}_2 \), the determinant only needs to be 1 to be invertible. This is essential in understanding which matrices in a set are functional and which are not.

Determinant Modulo Arithmetic

In matrix algebra, the determinant gives insights into the matrix's properties, including its invertibility. Modulo arithmetic, specifically modulo 2 arithmetic, simplifies each number to either 0 or 1.
In the group \( GL(2, \mathbb{Z}_2) \), determinants are calculated under these conditions where:

• Determinant calculation follows the same rule: \( ad - bc \).
• It is reduced by modulo 2, meaning possible outcomes for the determinant are 0 and 1.
• A determinant of 1 after modulo 2 arithmetic denotes an invertible matrix.

This arithmetic reduction aids in filtering out non-invertible matrices while retaining only those fit for inclusion in the general linear group.

Group Order

The order of a group in algebra relates to the number of elements it contains. When dealing with the general linear group \( GL(2, \mathbb{Z}_2) \), our task is to find all the unique matrices within it, specifically those that are invertible.
The step-by-step process to find the group order includes:

• First, determining possible matrices by considering every position's potential entries from \( \mathbb{Z}_2 = \{0, 1\} \).
• Then, calculating how many satisfy the invertibility (non-zero determinant).
• Finally, confirming through direct computation or known results, the group \( GL(2, \mathbb{Z}_2) \) consists of 6 matrices.

This concept allows us to identify how a group is structured and functions, essential for deeper group theory learning and application.

Matrix Algebra

Matrix algebra is a fundamental part of understanding systems and their transformations. In particular, it focuses on operations, properties, and applications of matrices.
In our scope:

• Matrices are arranged in rows and columns, forming a key component in many areas of mathematics.
• They support operations like addition, subtraction, multiplication, and inversion, integral in algebraic manipulations.
• In examining groups like \( GL(2, \mathbb{Z}_2) \), matrices show how transformations work on algebraic structures.

By mastering matrix algebra, learners grasp ways to model complex transformations and systems, making it indispensable for advanced study in mathematics and applied sciences.