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Magazine Chapter 1: Problem 21
What is the order of \(G L\left(2, \mathbb{Z}_{2}\right) ?\)
Short Answer
Expert verified
The order of \(GL(2, \mathbb{Z}_2)\) is 6.
Step by step solution
01
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\}.
02
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.
03
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 \(ad - bc \equiv 1 \pmod{2}\).
04
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.
05
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 \(GL(2, \mathbb{Z}_2)\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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:
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.
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:
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.
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:
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.
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:
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.
