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Chapter 9: Problem 33

Consider \(S_{n}\) for a fixed \(n \geq 2\) and let \(\sigma\) be a fixed odd permutation. Show that every odd permatation in \(S_{n}\) is a product of \(\sigma\) and some permutation in \(A_{n}\).

Short Answer

Expert verified
Any odd permutation in \( S_n \) can be expressed as \( \sigma \rho \) where \( \rho \) is in \( A_n \).

Step by step solution

01

Understanding the Problem

We are given an odd permutation \( \sigma \) in the symmetric group \( S_n \). Our goal is to express any odd permutation in \( S_n \) as a product of \( \sigma \) and a permutation from the alternating group \( A_n \). This means we need to show that for any odd permutation \( \tau \) in \( S_n \), there exists a permutation \( \rho \) in \( A_n \) such that \( \tau = \sigma \rho \).
02

Review Definitions and Properties

Recall that an odd permutation is one that can be expressed as a product of an odd number of transpositions (2-cycles), while a permutation in \( A_n \) is an even permutation (product of an even number of transpositions). Furthermore, every permutation in \( S_n \) can be written as a product of transpositions.
03

Expressing an Odd Permutation

Let \( \tau \) be any odd permutation in \( S_n \). Since \( \tau \) is odd, it can be expressed as a product of an odd number of transpositions: \( \tau = (t_1)(t_2)...(t_{2k+1}) \).
04

Using the Fixed Odd Permutation

Consider the permutation \( \rho = \sigma^{-1} \tau \). \( \sigma \) is odd, and \( \tau \) is odd, so the product \( \sigma^{-1} \tau \) will be even (since the composition of two odd permutations is even). Therefore, \( \rho \) belongs to \( A_n \).
05

Combining Permutations

Now, calculate \( \sigma \rho = \sigma (\sigma^{-1} \tau) = \tau \). This confirms that \( \tau \) can indeed be expressed as \( \sigma \rho \), where \( \rho \) is in \( A_n \). Therefore, every odd permutation in \( S_n \) is a product of \( \sigma \) and an even permutation, fulfilling the requirement of the problem.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Permutations
Permutations are simply the arrangements or rearrangements of elements in a set. It's a concept that illustrates how we can reorganize the order of a set.
  • An important aspect of permutations is their role in understanding symmetrical problems and structures.
  • For a set with 'n' elements, there are n! (n factorial) possible permutations.
In mathematics, permutations are used to see how various arrangements affect the operation or outcome. They can be expressed as a sequence of positions. In algebra, a permutation of a set is typically written in a cycle notation such as \( (1 ext{ }2 ext{ }3) \), indicating the permutation changes the first element to the second, the second to the third, and so forth, closing with the last changing to the first.
Symmetric Group
The symmetric group, denoted as \( S_n \), represents the set of all permutations of a finite set of 'n' elements. It's a cornerstone of group theory in mathematics as it embodies all possible ways to rearrange 'n' elements.
  • Each element in \( S_n \) is a permutation of the set.
  • It has a total of n! elements, as it includes all possible arrangements of the given set.
This group is significant in abstract algebra as it is the most general group capturing all symmetries within a set. Understanding the symmetric group is pivotal as it helps in analyzing symmetrical properties and invariant operations across different structures.
Alternating Group
The alternating group \( A_n \) is a subgroup of the symmetric group \( S_n \). It consists only of the even permutations of a set of 'n' elements. An even permutation is one that can be expressed as an even number of transpositions.
  • This group contains exactly half the number of elements as \( S_n \).
  • Its significance lies in its property of excluding all odd permutations, which provides a simple group structure to examine solely symmetrical even transformations.
The alternating group is particularly important in studies related to symmetry and symmetry breaking, where the focus is on those arrangements that maintain certain symmetrical properties without any imbalance caused by odd permutations.
Transpositions
Transpositions are the simplest form of permutation. They swap exactly two elements in a set, leaving all others unchanged. In cycle notation, a transposition is written as \((i ext{ }j)\), representing the interchange of these two positions.
  • They are the building blocks for all permutations, as any permutation can be expressed as a product of transpositions.
  • The parity of a permutation鈥攚hether it's odd or even鈥攊s determined by the number of transpositions of which it is a product.
Transpositions are vital because they provide an atomic level understanding of how permutations are formed, deconstructed, and influence the structure and order within various mathematical systems.

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Most popular questions from this chapter

In Exercises 1 through 6, find all orbits of the given permutation. $$ \sigma: \mathbb{Z} \rightarrow \mathbb{Z} \text { where } \sigma(n)=n+2 $$

In Exercises 20 through 22 , correct the definition of the italicized term without reference to the text, if correction is needed, so that it is in a form acceptable for pablication. The altemating group is the group of all even permutations.

In Exercises 7 through 9 , compute the indicated product of cycles that are permutations of \(\\{1,2,3,4,5,6,7,8\\}\). $$ (1,3,2,7)(4,8,6) $$

In Exercises 1 through 6, find all orbits of the given permutation. $$ \sigma: \mathbb{Z} \rightarrow \mathbb{Z} \text { where } \sigma(n)=n+1 $$

Mark each of the following true or false. a. Every permutation is a cycle. b. Every cycle is a permutation. c. The definition of even and odd permutations could have been given equally well before Theorem \(9.15\). d. Every nontrivial subgroup \(H\) of \(S_{9}\) containing some odd permutation contains a transposition. e. \(A_{5}\) has 120 elements. f. \(S_{n}\) is not cyclic for any \(n \geq 1\). g. \(A_{3}\) is a commatative group. h. \(S_{7}\) is isomorphic to the subgroup of all those elements of \(S_{8}\) that leave the number 8 fixed. I. \(S_{7}\) is isomorphic to the subgroup of all those elements of \(S_{n}\) that leave the number 5 fixed. 1\. The odd permutations in \(S_{8}\) form a subgroup of \(S_{8}\).
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