Show that the permutation group 𝑆ଷ is a group with respect to function composition. How many elements in 𝑆ଷ are of order 2? How many elements are self-invertible?

### 1. Showing that \( S_3 \) is a group with respect to function composition:

The **symmetric group** \( S_3 \) is the group of all permutations of three elements, say \( \{1, 2, 3\} \). To show that \( S_3 \) is a group with respect to function composition, we need to check if it satisfies the four group axioms:

- **Closure**: For any two elements \( \sigma \) and \( \tau \) in \( S_3 \), their composition \( \sigma \circ \tau \) must also be in \( S_3 \). Since function composition of permutations is another permutation, closure holds.

- **Associativity**: Function composition is associative by definition. That is, for any \( \sigma, \tau, \rho \in S_3 \), we have \( (\sigma \circ \tau) \circ \rho = \sigma \circ (\tau \circ \rho) \).

- **Identity element**: The identity element in \( S_3 \) is the identity permutation \( e = (1)(2)(3) \), which maps each element to itself. For any permutation \( \sigma \in S_3 \), we have \( \sigma \circ e = e \circ \sigma = \sigma \).

- **Inverses**: For every element \( \sigma \in S_3 \), there exists an inverse element \( \sigma^{-1} \in S_3 \) such that \( \sigma \circ \sigma^{-1} = \sigma^{-1} \circ \sigma = e \).

Thus, \( S_3 \) satisfies all the group axioms, and therefore it is a group under function composition.

### 2. Number of elements in \( S_3 \) of order 2:

The **order** of an element \( \sigma \in S_3 \) is the smallest integer \( k \) such that \( \sigma^k = e \), where \( e \) is the identity permutation.

The elements of \( S_3 \) are:
1. \( e = (1)(2)(3) \) — identity
2. \( (12) \) — transposition swapping 1 and 2
3. \( (13) \) — transposition swapping 1 and 3
4. \( (23) \) — transposition swapping 2 and 3
5. \( (123) \) — cyclic permutation: 1 → 2, 2 → 3, 3 → 1
6. \( (132) \) — cyclic permutation: 1 → 3, 3 → 2, 2 → 1

Now, we find the order of each element:
- \( e \): order 1 (since \( e^1 = e \)).
- \( (12), (13), (23) \): These are transpositions, and each of them squares to the identity: \( (12)^2 = e \), \( (13)^2 = e \), \( (23)^2 = e \). Hence, their order is 2.
- \( (123), (132) \): These are 3-cycles, and their order is 3 because \( (123)^3 = e \) and \( (132)^3 = e \).

Thus, there are **3 elements of order 2**: \( (12), (13), (23) \).

### 3. Number of self-invertible elements in \( S_3 \):

An element \( \sigma \in S_3 \) is **self-invertible** if \( \sigma = \sigma^{-1} \). This is true if \( \sigma^2 = e \). Therefore, the self-invertible elements are those of order 1 or 2.

- The identity \( e \) is trivially self-invertible.
- The elements \( (12), (13), (23) \), all of order 2, are also self-invertible.

Thus, there are **4 self-invertible elements** in \( S_3 \): \( e, (12), (13), (23) \).

### Summary:

- \( S_3 \) is a group under function composition.
- There are 3 elements in \( S_3 \) of order 2: \( (12), (13), (23) \).
- There are 4 self-invertible elements in \( S_3 \): \( e, (12), (13), (23) \).

Would you like further details or clarification?

Here are some related questions:
1. What is the general structure of \( S_n \), the symmetric group on \( n \) elements?
2. Can you explain why every transposition is of order 2 in any \( S_n \)?
3. How does the number of elements of each order vary as \( n \) increases in \( S_n \)?
4. What is the significance of self-invertible elements in group theory?
5. Can you describe the concept of conjugacy classes in \( S_3 \)?

**Tip**: In symmetric groups, transpositions are always of order 2, while cycles of length \( k \) have order \( k \).

Show that the permutation group 𝑆₃ is a group with respect to function composition. How many elements in 𝑆₃ are of order 2? How many elements are self-invertible?

Explore the properties of the symmetric group S₃, including proving it is a group under function composition. This detailed explanation covers the group axioms, identifies the elements of order 2, and determines how many elements are self-invertible. Suitable for undergraduate mathematics students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation