Problem 3 - 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. $$ p=x_{1} x_{2}+x_{2} x_{3}+x_{1} x_{3} $$

Step-by-Step Solution

Answer

The symmetries form the symmetric group \(S_3\) on \(x_1, x_2, x_3\).

Step 1: Understand the Task

We need to analyze the symmetries of the polynomial \(p = x_{1} x_{2} + x_{2} x_{3} + x_{1} x_{3}\). This involves finding the permutations of subscripts that leave the polynomial unchanged.

Step 2: Identify Permutable Subscripts

Consider the polynomial \(p = x_{1} x_{2} + x_{2} x_{3} + x_{1} x_{3}\). Notice that the terms are symmetric in \(x_{1}\), \(x_{2}\), and \(x_{3}\). The presence of each variable in every term suggests a degree of symmetry similar to the symmetric group.

Step 3: Determine Symmetric Permutations

The symmetry can be considered similar to that of the symmetric group \(S_3\) because the polynomial stays the same for permutations exchanging \(x_1\), \(x_2\), and \(x_3\). These permutations are:1. Identity: (1, 2, 3)2. Swap \(x_1\) and \(x_2\): (2, 1, 3)3. Swap \(x_2\) and \(x_3\): (1, 3, 2)4. Swap \(x_1\) and \(x_3\): (3, 2, 1)5. Circular shift: (2, 3, 1)6. Reverse circular shift: (3, 1, 2)

Step 4: Form the Group Table

Now, create the group table using the permutations. The table represents group operation, which is a composition of permutations: | | (1,2,3) | (2,1,3) | (1,3,2) | (3,2,1) | (2,3,1) | (3,1,2) | |-----|---------|---------|---------|---------|---------|---------| | (1,2,3) | (1,2,3) | (2,1,3) | (1,3,2) | (3,2,1) | (2,3,1) | (3,1,2) | | (2,1,3) | (2,1,3) | (1,2,3) | (3,2,1) | (1,3,2) | (3,1,2) | (2,3,1) | | (1,3,2) | (1,3,2) | (3,2,1) | (1,2,3) | (2,1,3) | (2,3,1) | (3,1,2) | | (3,2,1) | (3,2,1) | (1,3,2) | (2,1,3) | (1,2,3) | (3,1,2) | (2,3,1) | | (2,3,1) | (2,3,1) | (3,1,2) | (3,1,2) | (2,3,1) | (1,3,2) | (1,2,3) | | (3,1,2) | (3,1,2) | (2,3,1) | (2,3,1) | (3,1,2) | (1,2,3) | (1,3,2) | This table shows which permutation results when two permutations are combined.

Key Concepts

Polynomial Symmetries

Polynomials often exhibit fascinating characteristics when it comes to their symmetries. A polynomial is said to have a symmetry if it remains unaltered (unchanged) when certain operations, specifically permutations, are applied to its variables. When considering the polynomial \[ p = (x_1 - x_2)^2 + (x_3 - x_4)^2 \],the goal is to determine which permutations of its variables leave it unchanged.
The symmetries of such polynomials can be thought of as manipulating the subscripts while keeping the overall structure of the polynomial preserved. This means swapping certain indices does not alter the polynomial's value. These unchanged forms pinpoint the symmetrical nature of the polynomial, reflecting on the inherent balance and repeated pattern within its terms.Permutations leading to these symmetries can be grouped into mathematical entities known as symmetric groups. The purpose is to identify all permutations that maximize the preservation of polynomial form.

Permutations

In mathematics, a permutation refers to an ordered arrangement of elements from a set. When you have a polynomial like \[ p = x_1x_2 + x_2x_3 + x_1x_3 \], permutations of the variables can play a crucial role in identifying its symmetrical properties.
To explore this further, consider how you might rearrange the subscripts of the variables \(x_1, x_2, x_3\) while ensuring the polynomial's expression remains equivalent. Here:

• The identity permutation (1, 2, 3) involves no change, naturally preserving symmetry.
• Swapping \(x_1\) and \(x_2\) gives the permutation (2, 1, 3).
• Switching \(x_2\) and \(x_3\) results in (1, 3, 2).
• Changing \(x_1\) and \(x_3\) leads to the sequence (3, 2, 1).
• The circular and reverse circular shifts create permutations (2, 3, 1) and (3, 1, 2) respectively.

These permutations create a backdrop on which the symmetrical nature of the polynomial's structure can be studied more effectively. Each permutation offers a different yet equally valid arrangement that maintains the original value of the polynomial, thereby showcasing its symmetry.

Group Table

A group table is an essential tool in understanding the computational dynamics of permutations. It systematically chronicles the outcomes when you combine (or compose) different permutations of a set.
For the polynomial \[ p = x_1x_2 + x_2x_3 + x_1x_3 \], permutations like (1, 2, 3), (2, 1, 3), along with their others, can be arranged into a table to highlight their interactions. This table captures the closure property characteristic of groups, where combining any two permutations within the group results in another permutation from the same group.

• Start by listing permutations across the top and side headers of the table.
• Fill in the intersection cells with the resulting permutation from composing the row and column permutations.

Such a table provides a framework showing how the different symmetrical arrangements relate to one another. By tracing each combination, you ensure clarity on how these permutations interact, maintaining symmetry in the polynomial context. This tool is both a visual and structural insight into the deeper symmetries underlying polynomial representation.