Problem 2 - A Book of Abstract Algebra

Question

Write each of the following permutations in \(S_{9}\) as a product of disjoint cycles: (a) \(\left(\begin{array}{lllllllll}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ 4 & 9 & 2 & 5 & 1 & 7 & 6 & 8 & 3\end{array}\right)\) (b) \(\left(\begin{array}{lllllllll}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ 7 & 4 & 9 & 2 & 3 & 8 & 1 & 6 & 5\end{array}\right)\) (c) \(\left(\begin{array}{lllllllll}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ 7 & 9 & 5 & 3 & 1 & 2 & 4 & 8 & 6\end{array}\right)\) (d) \(\left(\begin{array}{ccccccccc}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ 9 & 8 & 7 & 4 & 3 & 6 & 5 & 1 & 2\end{array}\right)\)

Step-by-Step Solution

Answer

(a) \((1\ 4\ 5)(2\ 9\ 3)(6\ 7)(8)\), (b) \((1\ 7)(2\ 4)(3\ 9\ 5)(6\ 8)\), (c) \((1\ 7\ 4\ 3\ 5)(2\ 9\ 6)(8)\), (d) \((1\ 9\ 2\ 8)(3\ 7\ 5)(4)(6)\).

Step 1: Determine Cycles for Part (a)

Start from 1, which goes to 4, then 4 goes to 5, 5 goes to 1, forming the cycle \((1\ 4\ 5)\). Next, start from 2, which goes to 9, then 9 to 3, and finally 3 to 2, forming the cycle \((2\ 9\ 3)\). Number 6 goes to 7 and 7 returns to 6, forming the cycle \((6\ 7)\). Numbers 8 is unchanged, \((8)\). Thus, permutation as disjoint cycles is:\[(1\ 4\ 5)(2\ 9\ 3)(6\ 7)(8)\].

Step 2: Determine Cycles for Part (b)

Start from 1, which goes to 7, 7 goes to 1, forming the cycle \((1\ 7)\). Next, start from 2, which goes to 4, then 4 to 2, forming \((2\ 4)\). Number 3 goes to 9, 9 goes to 5, and 5 back to 3, forming the cycle \((3\ 9\ 5)\). Finally, 6 goes to 8, and 8 to 6, forming \((6\ 8)\). Number 2, \((2)\), is unchanged. Thus, permutation as disjoint cycles is:\[(1\ 7)(2\ 4)(3\ 9\ 5)(6\ 8)\].

Step 3: Determine Cycles for Part (c)

Start from 1, which goes to 7, then 7 to 4, and 4 back to 3, then 3 back to 5, 5 goes to 1, forming the single cycle \((1\ 7\ 4\ 3\ 5)\). Then, start from 2, which goes to 9, to 6, and finally 6 to 2, forming the cycle \((2\ 9\ 6)\). Number 8, \((8)\), is unchanged. Thus, permutation as disjoint cycles is:\[(1\ 7\ 4\ 3\ 5)(2\ 9\ 6)(8)\].

Step 4: Determine Cycles for Part (d)

Start from 1, which goes to 9, 9 to 2, and 2 back to 8, forming the cycle \((1\ 9\ 2\ 8)\). Start from 3, which goes to 7, 7 goes to 5, 5 to 3, forming the cycle \((3\ 7\ 5)\). Number 4, \((4)\), and 6, \((6)\), are unchanged. Thus, permutation as disjoint cycles is:\[(1\ 9\ 2\ 8)(3\ 7\ 5)(4)(6)\].

Key Concepts

Permutations

Permutations are arrangements or reorderings of a set of objects. In mathematics, particularly in group theory, permutations are crucial because they encapsulate the possible arrangements of elements in a set.
They are often represented by numbers that denote the positions to which the elements are moved. A simple way to visualize permutations is to imagine a set of numbered objects and how these can be swapped or shifted around.

• In the exercise above, each permutation is depicted as a rearrangement of the numbers 1 through 9.
• These numbers undergo a series of transitions that can be expressed as cycles.

Understanding permutations helps in solving complex problems in many areas like coding theory, combinatorics, and cryptography.

S_n

The notation \(S_n\) is often encountered in the realm of permutations and group theory. \(S_n\) represents the symmetric group of degree \(n\), which consists of all possible permutations of a set containing \(n\) elements.
For example, \(S_9\) includes all the permutations of the set \(\{1, 2, 3, \9\}\). These permutations are operations that rearrange the elements in all possible ways. Here are a few important points about \(S_n\):

• The total number of permutations in \(S_n\) is given by \(n!\) ("n factorial"), which is the product of all positive integers up to \(n\).
• Each element or permutation in \(S_n\) is invertible, meaning one can find an arrangement that returns all elements to their original order.

Understanding \(S_n\) is vital to solving exercises like the one given, where permutations in \(S_9\) are expressed as products of disjoint cycles.

Group Theory

Group theory is an area of abstract algebra that studies algebraic structures known as groups. In simple terms, a group is a set combined with an operation that satisfies certain conditions, such as associativity, identity, and invertibility.
Permutations are a foundational concept in group theory because the set of all permutations of a given set forms a symmetric group, \(S_n\).

• Each permutation can be considered as an element of a group operation known as composition.
• The exercise showcases how permutations from \(S_9\) can be decomposed into disjoint cycles, demonstrating the composition property.
• Group theory provides the tools to understand deeper properties of symmetry and structure in complex systems.

Studying permutations within the context of group theory allows one to explore the interplay between symmetry and algebraic operations in mathematics.

Step by Step Solution

Understanding how to convert permutations into disjoint cycles requires a structured approach, as demonstrated in the step-by-step solutions above. Let's break down this approach further:
Start by looking at each individual number or position in the permutation table. Trace where each number goes when the permutation is applied. This tracing step forms the basis for identifying cycles.

• A cycle represents a sequence where each number points to another one, cycling back to the starting number.
• Disjoint cycles don't share any common elements, simplifying the permutation into isolated paths.
• The process of converting a permutation into disjoint cycles facilitates a better understanding of the permutation's structure and behavior.

The step-by-step methodology ensures clarity and aids in visualizing movement transformations represented by permutations. It is a crucial skill for anyone delving into group theory or dealing with permutations.