Question

Mit \(G L(n, q)\) bezeichnen wir die Gruppe der invertierbaren \(\mathrm{n} \times \mathrm{n}\)-Matrizen mit Elementen aus \(G F(q)\). Bestimmen Sie die Ordnung von \(G L(n, q)\) für \(n=2\) und \(n=3\). Zusatzfrage: Können Sie eine allgemeine Formel für \(|G L(n, q)|\) angeben?

Short answer

For \(n=2\), \(|GL(2, q)| = q^2(q^2 - q)\); for \(n=3\), \(|GL(3, q)| = q^3(q^3 - q)(q^3 - q^2)\). The general formula is \(|GL(n, q)| = q^{n(n-1)/2} \\prod_{i=0}^{n-1} (q^n - q^i)\).

Step-by-step solution

Understanding the problem

We need to find the order of the general linear group \(GL(n, q)\) for matrices over the finite field \(GF(q)\) for \(n=2\) and \(n=3\). The task also asks for a general formula for \(|GL(n, q)|\).

Counting invertible matrices for \(n = 2\)

For \(n = 2\), we consider a 2x2 matrix with elements from \(GF(q)\). First, there are \(q^2\) choices for the first row. The second row must be linearly independent of the first; therefore, for the second row, we have \(q^2 - q\) choices (total \(q^2\) arrangements minus \(q\) multiples of the first row). Thus, the order of \(GL(2, q)\) is calculated as: \[ |GL(2, q)| = q^2(q^2 - q). \]

Calculating for \(n = 3\)

For \(n = 3\), the process is similar. The first row has \(q^3\) options. For the second row, linearly independent from the first, we have \(q^3 - q\) options. The third row must be independent of both previous rows. The number of choices for the third row is \(q^3 - q^2\). Therefore, the order \(|GL(3, q)|\) is: \[ |GL(3, q)| = q^3(q^3 - q)(q^3 - q^2). \]

Deriving the general formula

The general formula for the order of \(|GL(n, q)|\) involves choosing each row to be linearly independent of the ones above it. We start with \(q^n\) choices for the first row. For each subsequent row \(i\), we subtract the span of the previous \(i-1\) vectors. Thus, the formula is: \[ |GL(n, q)| = q^{n(n-1)/2} \prod_{i=0}^{n-1} (q^n - q^i). \]

Conclusion of calculation

The order of \(|GL(2, q)|\) is \(q^2(q^2 - q)\). The order of \(|GL(3, q)|\) is \(q^3(q^3 - q)(q^3 - q^2)\). The general formula for \(|GL(n, q)|\) is \(q^{n(n-1)/2} \prod_{i=0}^{n-1} (q^n - q^i)\).

Key concepts

Invertible Matrices

Invertible matrices are a fundamental concept in linear algebra. These matrices, also known as non-singular or non-degenerate matrices, possess an inverse. This means that for a matrix \(A\), an inverse matrix \(A^{-1}\) exists, such that the product of \(A\) and \(A^{-1}\) yields the identity matrix, denoted as \(I\). An identity matrix is a square matrix where all elements of the principal diagonal are ones, and all other elements are zeros.

To determine if a matrix is invertible, it must be square (having the same number of rows and columns) and its determinant must be non-zero. For example, if you have a 2x2 matrix \(A = \begin{pmatrix} a & b \ c & d \end{pmatrix}\), the matrix is invertible if \(ad - bc eq 0\). The determinant \(ad - bc\) is a scalar value giving information about the volume scaling factor of the linear transformation described by the matrix.

An important application of invertible matrices is in solving systems of linear equations, where finding the inverse can help determine the unique solution of the system.

Finite Field

A finite field, often denoted as \(GF(q)\) for Galois Field with \(q\) elements, is a set of finite numbers in which you can perform the usual operations of addition, subtraction, multiplication, and division (except by zero) and still stay within the field.

Finite fields are essential in algebra, particularly in relation to linear algebra and coding theory. They have the property that they contain a finite number of elements, making them useful in problems dealing with a discrete set of values. All finite fields are characterized by having a size that is a power of a prime number, \(q = p^n\), where \(p\) is a prime and \(n\) is a positive integer. For example, \(GF(2)\) is the simplest finite field containing the elements \(0\) and \(1\).

When dealing with a matrix over a finite field, each matrix element must be one of the field's elements. This plays a critical role in constructing general linear groups, as this restricts the potential values each matrix element can assume.

Order of a Group

In group theory, the order of a group is defined as the total number of elements within the group. For a finite group, this is simply a finite positive integer. Understanding the order of a group is crucial as it provides information about the group's structure and properties.

For example, with the general linear group \(GL(n, q)\), the order is determined by the number of invertible matrices possible for the given matrix size \(n\) over a finite field \(GF(q)\). We find the order by calculating how many different independent choices we can make for the rows of an \(n \times n\) matrix:

• First row: \(q^n\) independent possibilities
• Second row: must be linearly independent of the first, \(q^n - q\) possibilities
• Continue this pattern for further rows

Overall, the pattern leads to the general formula for \(|GL(n, q)|\) as: \[ |GL(n, q)| = q^{n(n-1)/2} \prod_{i=0}^{n-1} (q^n - q^i). \] This formula showcases the incremental restrictions applied as each new row must be linearly independent of the previous rows.

Linear Independence

Linear independence is a key concept in understanding vector spaces and matrix theory. A set of vectors is considered linearly independent if no vector in the set can be expressed as a linear combination of the others. In simple terms, you cannot derive any vector using any combination of the other vectors in the group. If such a derivation is possible, the set is linearly dependent.

This is essential in defining the rank of a matrix. The rank is the maximum number of linearly independent row or column vectors in the matrix. In the context of general linear groups, ensuring that rows of matrices are linearly independent guarantees the matrix's invertibility.

Practically, to determine if a set of vectors \(\{v_1, v_2, ..., v_n\}\) in a vector space is linearly independent, consider the equation \(c_1v_1 + c_2v_2 + ... + c_nv_n = 0\). The only solution should be \(c_1 = c_2 = ... = c_n = 0\). If this holds true, the vectors are linearly independent. Understanding this concept allows one to properly configure matrices to be part of general linear groups, as each row must be linearly independent of the others to ensure the matrix is non-singular.