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Chapter 7: Problem 1

Zeigen Sie: Es gibt genau zwei Permutationen aus \(S_{3}\), die Zyklen der L盲nge 3 sind; es gibt genau sechs Permutationen aus \(S_{4}\), die durch einen Zyklus der L盲nge 4 beschrieben werden. Allgemein: Es gibt genau \((n-1) !\) Permutationen aus \(S_{n}\), die durch einen Zyklus der L盲nge \(n\) beschrieben werden.

Short Answer

Expert verified
In \(S_3\), there are 2 three-cycles, in \(S_4\), there are 6 four-cycles; generally, in \(S_n\), there are \((n-1)!\) n-cycles.

Step by step solution

01

Understand the Problem

We are tasked with verifying the number of permutations that are cycles of specific lengths in symmetric groups. Specifically, we need to examine cycles of length 3 in \(S_3\) and cycles of length 4 in \(S_4\), and show that there are \((n-1)!\) cycles of length \(n\) in \(S_n\).
02

Examine Cycles in \(S_3\)

The symmetric group \(S_3\) consists of all permutations of the set \(\{1, 2, 3\}\). A 3-cycle permutation involves all three elements and cycles them. There are two such cycles: \((1 ext{ }2 ext{ }3)\) and \((1 ext{ }3 ext{ }2)\).
03

Verify the Count in \(S_3\)

For \(S_3\), there are \(3! = 6\) permutations in total. If we look for 3-cycles, once we fix the first element of the cycle (say 1), the remaining numbers can form two cycles \((1 ext{ }2 ext{ }3)\) and \((1 ext{ }3 ext{ }2)\). Thus, confirming that there are 2 such cycles in \(S_3\), consistent with \((3-1)! = 2\).
04

Examine Cycles in \(S_4\)

For the group \(S_4\), involving permutations of the set \(\{1, 2, 3, 4\}\), we seek 4-cycles. A 4-cycle uses all four numbers exactly once, such as \((1 ext{ }2 ext{ }3 ext{ }4)\).
05

Compute the Number of 4-Cycles

Each 4-cycle is uniquely determined by fixing the starting element and then ordering the remaining three elements. There are three choices for ordering these, so we have \((4-1)! = 6\) unique 4-cycles in \(S_4\).
06

Generalize to \(S_n\)

For a symmetric group \(S_n\), a complete cycle of length \(n\) is determined by selecting a sequence for \(n-1\) elements after fixing the first element. Therefore, there are \((n-1)!\) such cycles in \(S_n\), as it accounts for all possible orderings of the remaining elements.

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Key Concepts

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

Permutations
Permutations play a key role in the study of symmetric groups, which are collections of all possible orderings, or arrangements, of a set of elements. When you have a set with \( n \) elements, a permutation is simply a rearrangement of those \( n \) elements into a new sequence. For example, if you have the set \( \{ 1, 2, 3 \} \), the permutations include \( (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), \) and \( (3, 2, 1) \).
  • Permutations are fundamental in understanding how objects can be reorganized.
  • They form the basis for concepts in symmetric groups, which are covered in group theory.
By exploring permutations, you gain insights into possible sequences and structures within any given set.
Cycle Notation
Cycle notation is a compact way to express permutations. Instead of listing all elements in sequence, you describe how each element transitions to the next. This notation uses cycles, sequences that show how a group of elements maps onto itself. Consider the permutation of three elements \( (1, 2, 3) \) to \( (2, 3, 1) \). In cycle notation, this permutation is represented as \((1 \, 2 \, 3)\) meaning, 1 goes to 2, 2 goes to 3, and 3 goes back to 1.
  • Cycle notation greatly simplifies the understanding of permutations by highlighting the cyclic movement of elements.
  • Permutations in cycle notation can combine to form new permutations, showcasing their composite nature.
Mastering cycle notation is crucial for delving into more complex aspects of group theory.
Group Theory
Group theory is the mathematical study of symmetry through groups, sets equipped with operations satisfying specific principles like closure, associativity, identity, and invertibility. Symmetric groups, such as \( S_n \), are fundamental in this context, representing all possible permutations of \( n \) distinct elements.
  • Symmetric groups serve as a primary example of groups in mathematics and showcase all permutations of a specified set.
  • Understanding symmetric groups offers insights into broader applications across mathematics, physics, and chemistry.
By examining symmetric groups, you see patterns and symmetries manifest, making group theory an essential tool in solving complex algebraic problems.
Combinatorics
Combinatorics involves counting and arranging elements within a set, making it integral to understanding permutations and cycles in symmetric groups. It helps determine the number of possible permutations in a group, like how in \( S_3 \) there are \( 3! = 6 \) total permutations.
  • Combinatorial methods allow computation of how elements can be arranged or selected without reordering.
  • In symmetric groups, it helps in calculating the number of cycles of particular lengths, such as the \((n-1)!\) cycles of length \( n \).
Using combinatorics, you can systematically explore permutations and relationships between elements, enhancing problem-solving strategies in mathematics.

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Most popular questions from this chapter

Beweisen Sie die Entwicklung einer Determinante nach der ersten Spalte: Sei \(M=\left(m_{i j}\right)_{1 \leq i, j \leq n}\) eine Matrix aus \(K^{n \times n}\). Dann gilt: $$ \operatorname{det}(M)=\sum_{i=1}^{n}(-1)^{i+1} m_{i 1} \cdot \operatorname{det}\left(M_{i 1}\right) $$

Welche Determinanten haben die folgenden reellen Matrizen? $$ \left(\begin{array}{lll} 1 & 3 & 3 \\ 3 & 1 & 3 \\ 3 & 3 & 1 \end{array}\right),\left(\begin{array}{llll} 1 & 3 & 3 & 3 \\ 3 & 1 & 3 & 3 \\ 3 & 3 & 1 & 3 \\ 3 & 3 & 3 & 1 \end{array}\right),\left(\begin{array}{ccccc} 1 & a & 0 & \ldots & 0 \\ a & 1 & a & & \vdots \\ 0 & a & \ddots & \ddots & 0 \\ \vdots & & \ddots & 1 & a \\ 0 & \ldots & 0 & a & 1 \end{array}\right) $$

Zeigen Sie: Wenn \(A\) eine Elementarmatrix aus \(K^{n \times n}\) ist, bei der an der Stelle \((i, j)\) au脽erhalb der Hauptdiagonalen das Element \(a \neq 0\) steht, so ist \(A^{-1}\) diejenige Elementarmatrix, bei der an der Stelle \((i, j)\) das Element \(-a\) steht.

Berechnen Sie die Determinanten der folgenden reellen Matrizen: $$ \left(\begin{array}{ccc} -1 & 2 & 3 \\ 2 & 1 & 0 \\ 1 & 8 & 9 \end{array}\right),\left(\begin{array}{ccc} -1 & 0 & 5 \\ 0 & 0 & 7 \\ 3 & 3 & 3 \end{array}\right),\left(\begin{array}{cccc} 1 & 0 & 6 & 6 \\ 1 & 4 & 9 & 2 \\ 1 & 7 & 8 & 9 \\ 1 & 9 & 9 & 9 \end{array}\right) $$

Zeigen Sie, dass das Transponieren von Matrizen eine lineare Abbildung auf der Menge \(K^{n \times n}\) ist.
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