Permutation Computation And Sign Determination In S₅

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Introduction to Permutations

In the realm of abstract algebra, permutations play a crucial role, particularly within the symmetric group S_n, which consists of all possible permutations of n elements. A permutation is essentially a rearrangement of elements, and understanding them is fundamental in various mathematical contexts. This article delves into computing the product of permutations and expressing them in different forms, including as a product of transpositions. We will also explore how to determine the sign of a permutation, which is a vital concept in group theory.

Key Concepts in Permutations

Before diving into the specific problem, let's clarify some key concepts related to permutations:

  • Permutation: A permutation of a set is a bijective (one-to-one and onto) function from the set to itself. In simpler terms, it's a rearrangement of the elements of the set.
  • Cycle Notation: A concise way to represent permutations. For instance, (2 1 5) means that 2 is mapped to 1, 1 is mapped to 5, and 5 is mapped back to 2, forming a cycle. Elements not listed are assumed to be mapped to themselves.
  • Transposition: A permutation that swaps two elements and leaves the others unchanged. For example, (1 2) is a transposition that swaps 1 and 2.
  • Inverse of a Permutation: The permutation that reverses the effect of the original permutation. If β maps x to y, then β⁻¹ maps y back to x.
  • Sign of a Permutation: Every permutation can be expressed as a product of transpositions. The sign of a permutation is +1 if it can be written as a product of an even number of transpositions and -1 if it requires an odd number of transpositions. The sign is an important property that helps classify permutations as even or odd.

Understanding these concepts provides a solid foundation for tackling problems involving permutations.

Problem Statement: Computing σ = α · β⁻¹ in S₅

Given two permutations α = (2 1 5) and β = (2 3 5 4) in the symmetric group S₅, our task is threefold:

  1. Compute the permutation σ, which is the product of α and the inverse of β (β⁻¹).
  2. Express σ as a product of transpositions.
  3. Determine the sign of σ.

This problem encapsulates several important aspects of permutation manipulation, including cycle notation, inverse computation, permutation multiplication, and sign determination. Let's break down the solution step by step.

Step 1: Finding the Inverse of β (β⁻¹)

The first step in computing σ is to find the inverse of β. Given β = (2 3 5 4), we need to find a permutation that reverses the mapping of β. In cycle notation, β maps 2 to 3, 3 to 5, 5 to 4, and 4 back to 2. To find β⁻¹, we simply reverse these mappings.

β: 2 → 3, 3 → 5, 5 → 4, 4 → 2

To find β⁻¹, we reverse these mappings:

β⁻¹: 3 → 2, 5 → 3, 4 → 5, 2 → 4

Therefore, β⁻¹ = (2 4 5 3). This means that β⁻¹ maps 2 to 4, 4 to 5, 5 to 3, and 3 back to 2. Elements not included in the cycle (in this case, 1) are understood to be mapped to themselves.

Step 2: Computing σ = α · β⁻¹

Now that we have α = (2 1 5) and β⁻¹ = (2 4 5 3), we can compute σ = α · β⁻¹. This involves applying β⁻¹ first and then applying α. To do this, we trace the effect of each permutation on the elements 1 through 5.

Let's trace the mapping for each element:

  • 1: β⁻¹(1) = 1, then α(1) = 5. So, σ(1) = 5.
  • 2: β⁻¹(2) = 4, then α(4) = 4. So, σ(2) = 4.
  • 3: β⁻¹(3) = 2, then α(2) = 1. So, σ(3) = 1.
  • 4: β⁻¹(4) = 5, then α(5) = 2. So, σ(4) = 2.
  • 5: β⁻¹(5) = 3, then α(3) = 3. So, σ(5) = 3.

Thus, σ maps 1 to 5, 2 to 4, 3 to 1, 4 to 2, and 5 to 3. Writing this in cycle notation, we get σ = (1 5 3)(2 4).

This step demonstrates the process of multiplying permutations by tracing the movement of each element through the permutations.

Step 3: Expressing σ as a Product of Transpositions

To express σ as a product of transpositions, we need to break down each cycle into a series of two-element swaps. Recall that any cycle (a₁ a₂ ... aₖ) can be written as a product of transpositions:

(a₁ a₂ ... aₖ) = (a₁ aₖ)(a₁ aₖ₋₁) ... (a₁ a₂)

Applying this to σ = (1 5 3)(2 4):

  • (1 5 3) can be written as (1 3)(1 5)
  • (2 4) is already a transposition.

Therefore, σ = (1 5 3)(2 4) = (1 3)(1 5)(2 4). We have expressed σ as a product of three transpositions.

Step 4: Finding the Sign of σ

The sign of a permutation is determined by the parity (even or odd) of the number of transpositions in its decomposition. If a permutation can be written as a product of an even number of transpositions, its sign is +1 (even permutation). If it requires an odd number of transpositions, its sign is -1 (odd permutation).

In our case, σ = (1 3)(1 5)(2 4) is expressed as a product of three transpositions. Since three is an odd number, the sign of σ is -1.

Summary of the Solution

  1. We computed β⁻¹ = (2 4 5 3).
  2. We calculated σ = α · β⁻¹ = (1 5 3)(2 4).
  3. We expressed σ as a product of transpositions: σ = (1 3)(1 5)(2 4).
  4. We determined the sign of σ to be -1.

This comprehensive solution demonstrates how to manipulate permutations, compute their products and inverses, express them as transpositions, and determine their signs. These skills are crucial in abstract algebra and have applications in various fields, including cryptography and coding theory.

Conclusion

This article has provided a detailed walkthrough of computing the product of permutations, expressing them as transpositions, and determining their signs. By understanding these concepts and techniques, readers can tackle more complex problems in group theory and related areas. The example of computing σ = α · β⁻¹ in S₅ serves as a practical illustration of these principles, highlighting the importance of permutations in mathematics and its applications. The sign of permutation in this context serves as a crucial identifier, distinguishing between even and odd permutations, a concept with far-reaching implications in various mathematical domains. Understanding these computations and interpretations not only solidifies one's grasp of abstract algebra but also opens doors to more advanced topics and applications in diverse fields.