Question

Zeigen Sie, dass die Menge \(S L(n, q)\) der \(\mathrm{n} \times \mathrm{n}\)-Matrizen \(M\) über \(G F(q)\) mit \(\operatorname{det}(M)=1\) eine Untergruppe von \(G L(n, q)\) ist. Bestimmen Sie den Index dieser Untergruppe in \(G L(n, q)\)

Short answer

The set \( SL(n, q) \) is a subgroup of \( GL(n, q) \) with index \( q^{n-1}(q-1) \).

Step-by-step solution

Define the Groups

First, we need to understand the definitions: \( GL(n, q) \) is the group of all invertible \( n \times n \) matrices over the finite field \( GF(q) \). The group \( SL(n, q) \) consists of all matrices in \( GL(n, q) \) with determinant equal to 1.

Show Closure Under Multiplication

To prove that \( SL(n, q) \) is a subgroup of \( GL(n, q) \), we must show closure under multiplication. Suppose \( A, B \in SL(n, q) \). Then \( \det(A) = 1 \) and \( \det(B) = 1 \). We need \( AB \in SL(n, q) \), which requires \( \det(AB) = 1 \). Using the property \( \det(AB) = \det(A)\det(B) \), we get \( \det(AB) = 1 \times 1 = 1 \). Thus, \( AB \in SL(n, q) \).

Show Existence of Identity Element

The identity matrix \( I_n \) is in \( GL(n, q) \) and has determinant equal to 1. Therefore, \( I_n \in SL(n, q) \), satisfying the identity element requirement for \( SL(n, q) \).

Show Existence of Inverses

For a matrix \( A \in SL(n, q) \), \( \det(A) = 1 \). The inverse \( A^{-1} \) exists since \( A \in GL(n, q) \) and satisfies \( A A^{-1} = I_n \). We also have \( \det(A^{-1}) = (\det(A))^{-1} = 1^{-1} = 1 \). Therefore, \( A^{-1} \in SL(n, q) \).

Determine the Index

The index of \( SL(n, q) \) in \( GL(n, q) \) is the number of cosets of \( SL(n, q) \) in \( GL(n, q) \). Since \( GL(n, q) \) has \( q^n - 1 \) possible determinants (any non-zero element of \( GF(q) \)), the index is \( q^{n-1}(q-1) \) because the coset space can be represented by epochs with respect to each possible determinant.

Key concepts

Understanding Untergruppe

In algebra, a subgroup or "Untergruppe" in German is a smaller group contained within a larger group. To be considered a subgroup, this smaller group must satisfy certain conditions such as:

• Closure under multiplication: The product of any two elements in the subgroup must also be in the subgroup.
• Existence of an identity element: There must be an element in the subgroup that acts as the identity, not changing other elements when multiplied.
• Existence of inverses: Every element in the subgroup must have an inverse that is also in the subgroup.

In the context of Linear Algebra, when we say that a set like \( SL(n, q) \) is a subgroup, we are indicating that this set meets all the above criteria within the larger group \( GL(n, q) \). This guarantees \( SL(n, q) \) is well-defined in terms of operations inherited from \( GL(n, q) \).

The Role of Determinant

The determinant is a crucial concept in linear algebra, specifically when dealing with matrices. The determinant of a matrix provides vital information regarding the properties of the matrix, including:

• If a matrix is invertible. A non-zero determinant indicates that a matrix can be inverted.
• The scaling factor of area or volume when a matrix acts as a linear transformation.

To qualify as part of \( SL(n, q) \), a matrix must have a determinant of exactly 1. This stipulation ensures that, when viewed as a transformation, the matrix preserves volume, which in the context of finite fields, means maintaining the exact structure as it acts on space. Therefore, determinant acts as a filter to identify matrices with these specific properties necessary to form a subgroup.

Exploring Matrizen

Matrices, or "Matrizen" in German, are rectangular arrays of numbers that are fundamental objects in linear algebra. They are used to:

• Represent and solve systems of linear equations.
• Perform linear transformations on vector spaces.
• Encode data in fields such as computer graphics and machine learning.

In our discussion of \( SL(n, q) \) and \( GL(n, q) \), matrices serve as the building blocks of these groups. Each matrix in these groups not only provides a snapshot of potential linear transformations but also aligns with specific algebraic properties such as determinant and invertibility. Thus, understanding matrices is fundamental when exploring these group structures in more complex settings.

Introduction to Endlichen Körnern

The term "endlichen Körpern" refers to finite fields in algebra. A finite field is a set of numbers that contains a finite number of elements, used particularly over matrices where typical real-number operations still apply under modulo arithmetic.
A finite field, commonly denoted as \( GF(q) \), where \( q \) is a prime power, has the following properties:

• The field has exactly \( q \) elements.
• Fields support addition, subtraction, multiplication, and division by non-zero elements.

In the context of matrices, elements from these fields \( GF(q) \) are used to create the entries of \( n \times n \) matrices in \( GL(n, q) \) and \( SL(n, q) \). This modular approach allows us to work within known limits, which enables consistency and predictability when handling these algebraic objects.

Unraveling the Kosettengleichung

The term "Kosettengleichung" can be translated as "coset equation" in English. It is a key concept in group theory, especially when discussing subgroups. In essence, cosets are the partitioning of a group into subsets, where each subset holds:

• The same size as the subgroup used to form it.
• Elements derived by multiplying all elements of the subgroup by a fixed element from the main group.

Determining the index of a subgroup in a group involves counting the number of distinct cosets. For \( SL(n, q) \) within \( GL(n, q) \), the index tells us how many unique ways we can form cosets. The equation \( q^{n-1}(q-1) \), as given in the solution, reflects this partitioning and helps quantify the distinct separate "pieces" one can form, demonstrating the subgroup's coverage and overlap within the larger group's structure.