Problem 31 - Algebra: Pure and Applied

Question

Find all the elements in \(S_{4}\) of order \(2 .\)

Step-by-Step Solution

Answer

The elements of order 2 in \( S_4 \) are the transpositions: \((1 \, 2), (1 \, 3), (1 \, 4), (2 \, 3), (2 \, 4), (3 \, 4)\).

Step 1: Understand the Symmetric Group

The symmetric group, denoted as \( S_4 \), is the group of all permutations of 4 elements. It has \( 4! = 24 \) elements.

Step 2: Recognize Elements of Order 2

In the symmetric group \( S_4 \), an element has order 2 if it is a cycle of length 2 (a transposition). A transposition is a permutation that swaps two elements and leaves the others fixed.

Step 3: List All Transpositions

Identify all possible transpositions of the four elements 1, 2, 3, and 4. The transpositions are: \((1 \, 2), (1 \, 3), (1 \, 4), (2 \, 3), (2 \, 4), (3 \, 4)\).

Step 4: Confirm Each Element

Verify that each transposition indeed has an order of 2 by applying them twice: for a transposition \((a \, b)\), \(((a \, b))^2 = id\). Each transposition returns to the identity permutation after two applications, confirming they are of order 2.

Key Concepts

Permutations

Permutations are fundamental concepts in arranging objects in specific orders. In mathematics, a permutation of a set is a rearrangement of its elements. The symmetric group, denoted as \( S_n \) for a set with \( n \) elements, consists of all possible permutations of those elements.
For example, in the symmetric group \( S_4 \), you have 4 elements: \( 1, 2, 3, \) and \( 4 \). You can arrange these in \( 4! \) ways. This means there are 24 different permutations. Each permutation represents a unique way of ordering these elements. Some permutations are as simple as leaving the order the same (called the identity permutation), while others involve complex rearrangements.
Understanding permutations is crucial when working with symmetric groups since they form the basis of group operations. Recognizing how elements move and change positions can help in identifying permutations that fulfill specific conditions, such as having a certain order.

Transposition

A transposition is a specific type of permutation that swaps exactly two elements in a set while leaving all others in their original positions. Transpositions are visually simple but highly significant in the study of permutation groups because any permutation can be decomposed into a series of transpositions.

• In \( S_4 \), a group with four elements, a transposition takes two numbers and exchanges them, for instance, \((1 \, 2)\) swaps element 1 with element 2.
• This transposition can be expressed as a cycle notation \((a \, b)\).
• Each transposition directly modifies the positions of only two elements.

Transpositions are important because they are the building blocks of permutations. By applying a series of transpositions, one can achieve any permutation in the symmetric group. Each of these swaps is independent, which makes transpositions a straightforward concept yet powerful tool for analyzing permutations.

Order of Element

The order of an element in a group refers to the smallest number of times you must apply the element to return to the identity element (i.e., no change). It tells us how the element behaves under repetition.
For a permutation, particularly in the symmetric group \( S_4 \), the order of a transposition is 2. This means that if you perform a transposition twice, you return to the identity permutation.

• The identity element is the same as performing no operation on the elements, leaving them unchanged.
• For example, consider the transposition \((1 \, 2)\) in \( S_4 \): applying it once swaps 1 and 2. Applying it a second time swaps them back to their original places, resulting in the identity permutation.

Understanding the order of elements helps in characterizing the elements within a group. For the symmetric group, knowing the order of various permutations can further lead to insights into their properties and behaviors within the group structure.