Question

Machen Sie sich klar, dass \(\mathbf{Z}_{n}{ }^{*}\) bezüglich der Multiplikation modulo \(n\) eine Gruppe bildet. Machen Sie sich die Aussage zunächst am Beispiel \(n=15\) klar.

Short answer

The set \( \mathbf{Z}_{15}^{*} = \{1, 2, 4, 7, 8, 11, 13, 14\} \) forms a group under multiplication modulo \( 15 \).

Step-by-step solution

Define the Set

The set \( \mathbf{Z}_{15}^{*} \) consists of all integers from \( 1 \) to \( 14 \) that are coprime to \( 15 \). A number is coprime to \( 15 \) if its greatest common divisor (GCD) with \( 15 \) is \( 1 \).

Identify Coprime Numbers

Find the GCD of each number from \( 1 \) to \( 14 \) with \( 15 \). The numbers that are coprime to \( 15 \) are: \( 1, 2, 4, 7, 8, 11, 13, \) and \( 14 \). These numbers form the set \( \mathbf{Z}_{15}^{*} \).

Check Closure

Verify closure under multiplication modulo \( 15 \). Multiply each pair of elements in \( \mathbf{Z}_{15}^{*} \), and find their product modulo \( 15 \). Check if every result is also in \( \mathbf{Z}_{15}^{*} \). Example: \( 2 \times 4 = 8 \) and \( 8 \mod 15 = 8 \), which is in the set.

Verify Existence of Identity Element

The identity element in multiplication is \( 1 \). Since \( 1 \) is in \( \mathbf{Z}_{15}^{*} \) and satisfies \( a \times 1 \equiv a \mod 15 \) for all \( a \in \mathbf{Z}_{15}^{*} \), \( 1 \) is the identity element.

Check for Inverses

Check if every element has an inverse in \( \mathbf{Z}_{15}^{*} \). For each element \( a \), find an element \( b \) such that \( a \times b \equiv 1 \mod 15 \). Example: The inverse of \( 2 \) is \( 8 \) because \( 2 \times 8 = 16 \equiv 1 \mod 15 \).

Confirm Associativity

Multiplication is associative; thus, for any \( a, b, c \in \mathbf{Z}_{15}^{*} \), \( (a \times b) \times c = a \times (b \times c) \), ensuring associativity holds for the operation.

Key concepts

Z_n^*

The set \( \mathbf{Z}_{n}^{*} \) is a fundamental concept in group theory, particularly when we delve into groups formed under multiplication modulo \( n \). This set consists of all integers from \( 1 \) to \( n-1 \) that are coprime with \( n \). In easier terms, a number is considered coprime with another if their greatest common divisor (GCD) is 1. Since \( \mathbf{Z}_{n}^{*} \) contains elements that share this property, they become integral to forming a group structure with respect to modulo multiplication.
Understanding this requires checking each integer's GCD in the range up to \( n-1 \) to include only those that are coprime. Consequently, each element contributes to the unique identity of this group.
For instance, \( \mathbf{Z}_{15}^{*} \) includes the integers \( 1, 2, 4, 7, 8, 11, 13, \text{ and } 14 \) since these numbers are coprime with 15, ultimately allowing us to explore modular arithmetic within the framework provided by group theory.

Modulo-Multiplikation

Modulo-multiplication is a crucial operation in number theory, forming the backbone of the arithmetic group concept with \( \mathbf{Z}_{n}^{*} \). When we say multiplication modulo \( n \), it means that when we multiply two numbers, we take the remainder when divided by \( n \). For example, multiplying \( 8 \) by \( 11 \) yields \( 88 \), and when divided by 15, the remainder is 13, hence \( 88 \equiv 13 \mod 15 \).
This operation has several distinct properties within \( \mathbf{Z}_{n}^{*} \):

• **Closure**: Whenever two numbers from \( \mathbf{Z}_{n}^{*} \) are multiplied, the result, when reduced by modulo \( n \), is always another element within the set.
• **Existence of Identity Element**: The identity element for multiplication is \( 1 \) as any number multiplied by \( 1 \) remains unchanged within the modulo operation.
• **Inverses**: Every element in \( \mathbf{Z}_{n}^{*} \) has a multiplicative inverse in the same set, ensuring the product is \( 1 \).
• **Associativity**: Similar to regular multiplication, it is both associative and consistent under modulo arithmetic.

These features facilitate constructing a group under modulo multiplication, reinforcing the interaction of arithmetic properties.

Zahlentheorie

Number theory, or **Zahlentheorie** in German, forms the mathematical foundation for exploring properties and relationships of numbers, particularly integers. A key aspect within this is understanding divisibility and prime numbers, which play a vital role in defining coprime numbers and constructing the set \( \mathbf{Z}_{n}^{*} \).
In our focus on group structures such as \( \mathbf{Z}_{15}^{*} \), number theory helps in comprehending how these constructs behave. Using the Euclidean algorithm, for instance, empowers us to determine the greatest common divisor (GCD) effectively, confirming elements are coprime.
Number theory isn't just theoretical. Its practical applications span cryptographic systems, data security, and creating efficient algorithms. Therefore, appreciating its principles extends beyond academic exercises, influencing many technological advancements. The beauty of Zahlentheorie lies in its elegance and applicability in solving real-world problems.

Koprime Zahlen

Coprime numbers are two integers whose greatest common divisor (GCD) is 1. This fundamental concept is vital in the classification of elements within \( \mathbf{Z}_{n}^{*} \). When constructing sets like \( \mathbf{Z}_{15}^{*} \), identifying which integers between \( 1 \) and \( 14 \) are coprime with 15 is crucial.
To determine if two numbers are coprime, we use the Euclidean algorithm, a straightforward technique to find their GCD. For example, \( 7 \) and \( 15 \) are coprime, verified by tracing the division steps to confirm a GCD of 1.
In modular arithmetic, having coprime numbers ensures that multiplicative inverses exist, which is critical for solving equations in modular systems. Applications include cryptographic protocols where security relies on the properties of coprime numbers.
Without understanding coprime numbers, one cannot fully appreciate number theory's breadth or effectively explore modular arithmetic's elegance. Recognizing the significance of these numbers opens doors to a deeper understanding of mathematical group structures and their applications.