Problem 2 - A Book of Abstract Algebra

Question

Consider the polynomial \(p=\left(x_{1}-x_{2}\right)^{2}+\left(x_{3}-x_{4}\right)^{2}\). It is unaltered when the subscripts undergo any of the following permutations: $$ \begin{array}{lll} \left(\begin{array}{llll} 1 & 2 & 3 & 4 \\ 2 & 1 & 3 & 4 \end{array}\right) & \left(\begin{array}{llll} 1 & 2 & 3 & 4 \\ 1 & 2 & 4 & 3 \end{array}\right) & \left(\begin{array}{llll} 1 & 2 & 3 & 4 \\ 2 & 1 & 4 & 3 \end{array}\right) & \left(\begin{array}{llll} 1 & 2 & 3 & 4 \\ 3 & 4 & 1 & 2 \end{array}\right) \\ \left(\begin{array}{llll} 1 & 2 & 3 & 4 \\ 4 & 3 & 1 & 2 \end{array}\right) & \left(\begin{array}{llll} 1 & 2 & 3 & 4 \\ 3 & 4 & 2 & 1 \end{array}\right) & \left(\begin{array}{llll} 1 & 2 & 3 & 4 \\ 4 & 3 & 2 & 1 \end{array}\right) & \left(\begin{array}{llll} 1 & 2 & 3 & 4 \\ 1 & 2 & 3 & 4 \end{array}\right) \end{array} $$For example, the first of these permutations replaces \(p\) by $$ \left(x_{2}-x_{1}\right)^{2}+\left(x_{3}-x_{4}\right)^{2} $$ the second permutation replaces \(p\) by \(\left(x_{1}-x_{2}\right)^{2}+\left(x_{4}-x_{3}\right)^{2} ;\) and so on. The symmetries of a polynomial \(p\) are all the permutations of the subscripts which leave \(p\) unchanged. They form a group of permutations. List the symmetries of each of the following polynomials, and write their group table. $$ p=\left(x_{1}-x_{2}\right)\left(x_{2}-x_{3}\right)\left(x_{1}-x_{3}\right) $$

Step-by-Step Solution

Answer

The symmetries of the polynomial form the symmetric group \( S_3 \).

Step 1: Understand the Permutations of the First Polynomial

The problem statement describes the polynomial \( p=(x_1-x_2)^2+(x_3-x_4)^2 \), which remains unchanged when certain permutations are applied. These permutations are essentially rearrangements of the indices in such a way that they don't alter the polynomial. These permutations include swapping \( x_1 \) with \( x_2 \), \( x_3 \) with \( x_4 \), switching the pairs \((x_1, x_2)\) with \( (x_3, x_4) \), and a few others. These permutations keep the polynomial expression identical.

Step 2: Define the New Polynomial to Explore

The polynomial to analyze is \( p'=(x_1-x_2)(x_2-x_3)(x_1-x_3) \). We need to determine which permutations of the indices \( 1, 2, 3 \) leave this polynomial unchanged.

Step 3: Identify Symmetries in the New Polynomial

Consider how each permutation might affect \( p' \). \( p' \) is unchanged if swapping indices results in no alteration of the resulting polynomial. For example, swapping \( x_1 \) and \( x_2 \) turns \( (x_1-x_2)(x_2-x_3)(x_1-x_3) \) into \( (x_2-x_1)(x_1-x_3)(x_2-x_3) \), which results in the same product. The permutations that leave \( p' \) unchanged can include \((1,2,3)\), \((1,3,2)\), \((2,1,3)\), and others.

Step 4: List the Symmetries

Upon analysis, the set of permutations that leave the polynomial unchanged form a group of symmetries. These are essentially the identities, the swaps of pairs, and cycles among them: 1. \((1,2,3)\), 2. \((1,3,2)\), 3. \((2,1,3)\), 4. \((2,3,1)\), 5. \((3,1,2)\), 6. \((3,2,1)\). This matches the symmetric group \( S_3 \) which comprises all permutations of three elements.

Step 5: Write the Group Table

Create the group table for the symmetries of the polynomial \( p'=(x_1-x_2)(x_2-x_3)(x_1-x_3) \). List all permutations that form the group and construct a Cayley table to show how they combine:Columns and rows are labeled with permutations: Id, \((1 2)\), \((1 3)\), \((2 3)\), \((1 2 3)\), \((1 3 2)\).Fill the table with the result of composing each pair of permutations, considering their outcomes.This forms a typical symmetric group \( S_3 \) table with entries showing how permutations combine and result in one another.

Key Concepts

Symmetric Group

In mathematics, the concept of a symmetric group plays a vital role in understanding permutations. A symmetric group, denoted as \( S_n \), is the group of all possible permutations of \( n \) elements. The elements of \( S_n \) are the different ways of arranging \( n \) objects, each arrangement being a different permutation.

Symmetric groups are fundamental in algebra because they highlight the symmetry properties in a set and allow us to analyze permutations in a structured manner. For instance, in our polynomial exercise, we look at specific permutations that do not alter the value of the polynomial. This set of permutations forms a subgroup of the symmetric group.

Each permutation in the symmetric group can be written as a product of transpositions (simple switches of two elements), indicating that all permutations are built from these basic operations. The symmetric group has special properties, such as being closed under the operation of composition and having an identity element, making it essential for various applications in mathematics.

Polynomial Symmetries

Polynomial symmetries refer to the permutations of variables that leave a polynomial unchanged. In simple terms, when you rearrange the variables of a polynomial and if the polynomial looks the same, then the rearrangement is a symmetry.

Take the polynomial \( p = (x_1 - x_2)^2 + (x_3 - x_4)^2 \); its symmetries include permutations such as swapping \( x_1 \) with \( x_2 \) and \( x_3 \) with \( x_4 \). These operations do not affect the polynomial's form, showcasing its symmetry.

Understanding polynomial symmetries is crucial because they help in identifying underlying patterns and invariants within mathematical expressions. They can also simplify polynomial equations by reducing the number of distinct forms to consider. When these symmetries are collected, they form a group, showcasing how permutations can systematically rearrange parts without altering the overall structure.

Cayley Table

A Cayley table is a useful way to visualize and organize how elements within a group interact with each other when combined. Named after the mathematician Arthur Cayley, it's primarily used in group theory to describe the product of two group elements.

The table is structured with rows and columns labeled by group elements. The intersection of a row and a column in the table gives the product of those two elements. In our polynomial example, where the symmetric group \( S_3 \) is considered, a Cayley table shows how various permutations combine.

Creating a Cayley table for a group like \( S_3 \) involves listing all permutations of three elements and showing how they compose with one another. This helps in identifying properties such as identity elements and inverse relationships. The Cayley table is a handy tool for understanding the structure and operation of a group, offering a clear and systematic way of representing group interactions.