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Chapter 9: Problem 21

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 publication. A cycle is a permatation having only one orblt.

Short Answer

Expert verified
A cycle is a permutation with only one orbit.

Step by step solution

01

Identify the Incorrect Term

The term "permatation" seems to be a typographical error. The correct term should be "permutation." A permutation is an arrangement or rearrangement of the elements of a set.
02

Define Permutation

A permutation is a specific arrangement or sequence of elements from a set, where the order of elements is significant. For example, the permutations of the set \( \{1, 2, 3\} \) include \([1, 2, 3]\), \([1, 3, 2]\), and so on.
03

Define Orbit in Context of Cycle

In the context of permutations and group theory, an orbit is a set of elements that are permuted among themselves within a permutation. A permutation can thus be decomposed into disjoint cycles that describe these orbits.
04

Correct Definition of Cycle

A cycle in the context of permutations is a sequence of elements of a set where each element is replaced by the next one in the sequence, and the last one by the first element, forming a closed loop. This corresponds to having exactly one orbit in the permutation, as all elements involved are cycled through in a singular sequence.
05

Correct the Sentence

Replace the original statement with the correct definition: "A cycle is a permutation that contains only one orbit." This accurately describes the concept where a cycle involves a single continuous loop through the elements.

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Key Concepts

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

Permutation
In mathematics, particularly in the field of group theory, a permutation represents a reordering of elements within a set. This is not simply moving items around; it involves specific sequences that change the positions of the elements. Understanding permutations is crucial because they form the backbone of much of modern algebra.
  • Definition: A permutation of a set is any reordering of its elements. For example, consider a set: \( \{1, 2, 3\} \). The possible permutations include \([1, 2, 3]\), \([3, 2, 1]\), \([2, 1, 3]\), and so on. These different orderings are each a different permutation.

  • Importance in Group Theory: In group theory, permutations are studied to understand symmetrical properties and transformations of structures. Groups themselves can be thought of as sets equipped with a permutation operation, where this operation is associative and reversible.
Permutations play a vital role across different branches of mathematics and computer science, often used in algorithms and to solve complex combinatorial problems.
Orbit
The concept of an orbit in group theory and permutations is central to understanding how elements within a permutation group are related. When considering permutations as transformations, each element of a set will cycle through a series of transformations, which is termed its orbit.
  • Definition: An orbit is the set of all possible positions that a single element can occupy after applying the permutations of the group. In simpler terms, it's the path or positions an element can traverse under the action of a specific permutation.

  • Relation to Cycle: In the context of cycles, an orbit is simply the collection of elements involved in the cycle. For instance, if a cycle within a permutation group transforms element \(a\) to \(b\), \(b\) to \(c\), and \(c\) back to \(a\), the orbit of this cycle is \( \{a, b, c\} \).
Understanding orbits helps in visualizing and computing the sequences and transformations in permutation groups, aiding significantly in problem-solving and theoretical interpretations.
Cycle
A cycle in permutation is one of the simplest yet most powerful concepts within the study of permutations and group theory. Cycles provide a compact way to understand how elements in a set are permuted, showing the closed loop path of an element through each transformation.
  • Definition: A cycle within a permutation is a sequence where each element replaces the next, and the last element reverts to the first. Essentially, it describes a closed loop or singular orbit, making cycles particularly easy to analyze and write.

  • Example: Consider a permutation of the set \( \{1, 2, 3, 4 \} \) where the transformation path is: \( (1 \to 2), (2 \to 4), (4 \to 1) \). This forms a cycle \((1, 2, 4)\), and 3 is unchanged, forming a cycle itself: \((3)\).
Cycles simplify the description of complex permutations and make it easier to perform calculations, such as computing the order of permutations or analyzing symmetry properties.

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Most popular questions from this chapter

In Exercises 10 through 12, express the permutation of \(\\{1,2,3,4,5,6,7,8\\}\) as a product of disjoint cycles, and then as a product of transposithons \(\left(\begin{array}{llllllll}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ 3 & 6 & 4 & 1 & 8 & 2 & 5 & 7\end{array}\right)\)

In Exercises 1 through 6, find all orbits of the given permutation. $$ \left(\begin{array}{llllll} 1 & 2 & 3 & 4 & 5 & 6 \\ 5 & 1 & 3 & 6 & 2 & 4 \end{array}\right) $$

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 14 through 18 , find the maximum possible order for an element of \(S_{n}\) for the given value of \(n\). n=10

Prove the following about \(S_{n}\) if \(n \geq 3\). a. Every permutation in \(S_{n}\) can be written as a product of at most \(n-1\) transpositions. b. Every permutation in \(S_{n}\) that is not a cycle can be written as a product of at most \(n-2\) transpositions. c. Every odd permatation in \(S_{n}\) can be written as a product of \(2 n+3\) transpositions, and every even permutation as a product of \(2 n+8\) transpositions.
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