Question

Let \(G\) be a group of order 15 . Show that \(G\) contains an element of order \(3 .\)

Short answer

Group \(G\) contains an element of order 3, by Cauchy's Theorem.

Step-by-step solution

Prime Factorization of Group Order

The order of the group \(G\) is given as 15. First, determine the prime factorization of 15. We have 15 = 3 \times 5.

Apply Cauchy's Theorem

Cauchy's Theorem for groups states that if a prime number \(p\) divides the order of a group, then the group contains an element of order \(p\). Since 3 is a prime factor of 15, by Cauchy's Theorem, \(G\) must contain an element of order 3.

Conclude Existence of Element

Since the prime \(3\) divides the order of the group \(G\), verify that Cauchy's Theorem applies: indeed, \(G\) contains at least one element whose order is 3.

Key concepts

Cauchy's Theorem

Cauchy's Theorem is a fundamental result in group theory that provides insights into the structure of groups. It states that if a prime number \(p\) divides the order of a finite group \(G\), then there exists at least one element in \(G\) that has an order exactly equal to \(p\). This theorem helps in understanding the internal makeup of a group by pointing out the presence of certain elements based on divisibility.

For example, in a group of order \(15\), which breaks down into prime factors as \(15 = 3 \times 5\), both \(3\) and \(5\) are primes dividing the order of the group. According to Cauchy's Theorem, \(G\) must contain elements of order \(3\) and \(5\). This discovery can tremendously simplify solving group-related problems by confirming the existence of elements with specific orders.

Prime Factorization

Prime factorization is the process of breaking down a composite number into a product of its prime numbers. In group theory, understanding the prime factorization of a group's order is crucial, especially when using Cauchy's Theorem.

Consider the group \(G\) with an order of \(15\). The prime factors are found by dividing the group order into its basic prime constituents: \(3\) and \(5\). These prime numbers reveal the potential existence of elements with matching orders within the group.

By revealing the underlying structure of a group's order, prime factorization helps determine how many elements within the group can have certain properties or behaviors. This is particularly useful in determining the application of theorems that rely on prime divisors.

Group Order

The group order is a fundamental concept in group theory, denoting the total number of elements within a group \(G\). It provides a broad view of the group's size and serves as a starting point for deeper analyses.

Knowing the order of a group can immediately suggest strategies for exploring the group's characteristics. For example, a group with order \(15\) suggests the potential exploration of its subgroups and elements, as it can be broken down into its prime factors \(15 = 3 \times 5\).

The order gives clues about the presence of certain kinds of elements and the application of theorems like Cauchy's, directing us toward concluding details about these structures efficiently.

Elements of a Group

Elements of a group are the fundamental building blocks of group theory. Each element represents a member of the group, and together they exhibit the group's structure and properties.

When considering elements within a particular group, like one of order \(15\), key concepts such as element order and identity are crucial. Group theory examines how these elements combine under the group's operation to yield identities and inverses.

The order of an element refers to the smallest positive number \(n\) such that raising the element to the \(n\)th power returns the identity element. For instance, in our example, the presence of elements of order \(3\) is assured by the prime factorization and Cauchy's Theorem, giving insight into the dynamic interplay within the group.