Contemporary Abstract Algebra

Joseph Gallian

Chapter 5 Permutation Groups - all with Video Answers

Educators

Chapter Questions

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Problem 1

Let
$$
\alpha=\left[\begin{array}{llllll}
1 & 2 & 3 & 4 & 5 & 6 \\
2 & 1 & 3 & 5 & 4 & 6
\end{array}\right] \text { and } \beta=\left[\begin{array}{llllll}
1 & 2 & 3 & 4 & 5 & 6 \\
6 & 1 & 2 & 4 & 3 & 5
\end{array}\right]
$$
Compute each of the following.
a. $\alpha$
b. $\beta \alpha$
c. $\alpha \beta$

Victor Salazar

Numerade Educator

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Problem 2

$$
\alpha=\left[\begin{array}{llllllll}
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\
2 & 3 & 4 & 5 & 1 & 7 & 8 & 6
\end{array}\right] \text { and } \beta=\left[\begin{array}{llllllll}
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\
1 & 3 & 8 & 7 & 6 & 5 & 2 & 4
\end{array}\right] \text { . }
$$
Write $\alpha, \beta$, and $\alpha \beta$ as
a. products of disjoint cycles;
b. products of 2 -cycles.

Victor Salazar

Numerade Educator

03:16

Problem 3

Write each of the following permutations as a product of disjoint cycles.
a. $(1235)(413)$
b. $(13256)(23)(46512)$
c. $(12)(13)(23)(142)$

Clarissa Noh

Numerade Educator

02:52

Problem 4

Find the order of each of the following permutations.
a. (14)
b. (147)
c. (14762)
d. $\left(a_{1} a_{2} \cdots a_{k}\right)$

James Chok

Numerade Educator

01:47

Problem 5

What is the order of each of the following permutations?
a. (124)(357)
b. $(124)(3567)$
c. $(124)(35)$
d. (124)(357869)
e. $(1235)(24567)$
f. $(345)(245)$

Taha T

Numerade Educator

01:47

Problem 6

What is the order of each of the following permutations?
a. $\left[\begin{array}{llllll}1 & 2 & 3 & 4 & 5 & 6 \\ 2 & 1 & 5 & 4 & 6 & 3\end{array}\right]$
b. $\left[\begin{array}{lllllll}1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 7 & 6 & 1 & 2 & 3 & 4 & 5\end{array}\right]$

Taha T

Numerade Educator

00:22

Problem 7

What is the order of the product of a pair of disjoint cycles of lengths 4 and 6 ?

James Kiss

Numerade Educator

01:24

Problem 8

Determine whether the following permutations are even or odd.
a. $(135)$
b. $(1356)$
c. (13567)
d. $(12)(134)(152)$
e. $(1243)(3521)$

Prabhu Ramji

Numerade Educator

00:46

Problem 9

What are the possible orders for the elements of $S_{6}$ and $A_{6}$ ? What about $A_{7} ?$ (This exercise is referred to in Chapter 25.)

Tanishq Gupta

Numerade Educator

01:12

Problem 10

Show that $A_{8}$ contains an element of order 15 .

Tanishq Gupta

Numerade Educator

01:42

Problem 11

Find an element in $A_{12}$ of order 30 .

Nick Johnson

Numerade Educator

02:26

Problem 12

Show that a function from a finite set $S$ to itself is one-to-one if and only if it is onto. Is this true when $S$ is infinite? (This exercise is referred to in Chapter 6.)

Adam Dehollander

Numerade Educator

00:45

Problem 13

Suppose that $\alpha$ is a mapping from a set $S$ to itself and $\alpha(\alpha(x))=x$ for all $x$ in $S$. Prove that $\alpha$ is one-to-one and onto.

James Kiss

Numerade Educator

01:17

Problem 14

Suppose that $\alpha$ is a 6-cycle and $\beta$ is a 5 -cycle. Determine whether $\alpha^{5} \beta^{4} \alpha^{-1} \beta^{-3} \alpha^{5}$ is even or odd. Show your reasoning.

Uma Kumari

Numerade Educator

01:32

Problem 15

Let $n$ be a positive integer. If $n$ is odd, is an $n$ -cycle an odd or an even permutation? If $n$ is even, is an $n$ -cycle an odd or an even permutation?

William Mead

Numerade Educator

02:00

Problem 16

If $\alpha$ is even, prove that $\alpha^{-1}$ is even. If $\alpha$ is odd, prove that $\alpha^{-1}$ is odd.

Adriano Chikande

Numerade Educator

00:45

Problem 17

Prove Theorem $5.6$.

Amrita Bhasin

Numerade Educator

01:03

Problem 18

In $S_{n}$, let $\alpha$ be an $r$ -cycle, $\beta$ an $s$ -cycle, and $\gamma$ a $t$ -cycle. Complete the following statements: $\alpha \beta$ is even if and only if $r+s$ is $\ldots ; \alpha \beta \gamma$ is even if and only if $r+s+t$ is $\ldots$

Hast Aggarwal

Numerade Educator

01:41

Problem 19

Let $\alpha$ and $\beta$ belong to $S_{n}$. Prove that $\alpha \beta$ is even if and only if $\alpha$ and $\beta$ are both even or both odd.

Adriano Chikande

Numerade Educator

11:02

Problem 20

Associate an even permutation with the number $+1$ and an odd permutation with the number $-1$. Draw an analogy between the result of multiplying two permutations and the result of multiplying their corresponding numbers $+1$ or $-1$.

Sirat Shah

Numerade Educator

00:33

Problem 21

Complete the following statement: A product of disjoint cycles is even if and only if

Ashley Volpe

Numerade Educator

02:48

Problem 22

What cycle is $\left(a_{1} a_{2} \cdots a_{n}\right)^{-1}$ ?

Gregory Higby

Numerade Educator

01:31

Problem 23

Show that if $H$ is a subgroup of $S_{n}$, then either every member of $H$ is an even permutation or exactly half of the members are even. (This exercise is referred to in Chapter 25.)

Varsha Aggarwal

Numerade Educator

00:59

Problem 24

Suppose that $H$ is a subgroup of $S_{n}$ of odd order. Prove that $H$ is a subgroup of $A_{n}$

Varsha Aggarwal

Numerade Educator

04:18

Problem 25

Give two reasons why the set of odd permutations in $S_{n}$ is not a subgroup.

Sirat Shah

Numerade Educator

03:06

Problem 26

Let $\alpha$ and $\beta$ belong to $S_{n}$. Prove that $\alpha^{-1} \beta^{-1} \alpha \beta$ is an even permutation.

Sriparna Bhattacharjee

Numerade Educator

03:24

Problem 27

How many elements are there of order 2 in $A_{8}$ that have the disjoint cycle form $\left(a_{1} a_{2}\right)\left(a_{3} a_{4}\right)\left(a_{5} a_{6}\right)\left(a_{7} a_{8}\right) ?$

Niamat Khuda

Numerade Educator

00:39

Problem 28

How many elements of order 5 are in $S_{7}$ ?

Rajpal Sian

Numerade Educator

01:25

Problem 29

How many elements of order 4 does $S_{6}$ have? How many elements of order 2 does $S_{6}$ have?

Gregory Higby

Numerade Educator

03:18

Problem 30

Prove that (1234) is not the product of 3 -cycles. Generalize.

Carlos Pinilla

Numerade Educator

01:24

Problem 31

Let $\beta \in S_{7}$ and suppose $\beta^{4}=(2143567)$. Find $\beta$. What are the possibilities for $\beta$ if $\beta \in S_{9} ?$

Aman Gupta

Numerade Educator

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Problem 32

Let $\beta=(123)(145)$. Write $\beta^{99}$ in disjoint cycle form.

Victor Salazar

Numerade Educator

01:13

Problem 33

Let $\left(a_{1} a_{2} a_{3} a_{4}\right)$ and $\left(a_{5} a_{6}\right)$ be disjoint cycles in $S_{10}$. Show that there is no element $x$ in $S_{10}$ such that $x^{2}=\left(a_{1} a_{2} a_{3} a_{4}\right)\left(a_{5} a_{6}\right)$.

Hoan Nguyen

Numerade Educator

00:26

Problem 34

If $\alpha$ and $\beta$ are distinct 2 -cycles, what are the possibilities for $|\alpha \beta|$ ?

Sam Limsuwannarot

Numerade Educator

01:40

Problem 35

Let $G$ be a group of permutations on a set $X$. Let $a \in X$ and define $\operatorname{stab}(a)=\{\alpha \in G \mid \alpha(a)=a\} .$ We call stab $(a)$ the stabilizer of a in $G$ (since it consists of all members of $G$ that leave $a$ fixed). Prove that $\operatorname{stab}(a)$ is a subgroup of $G$. (This subgroup was introduced by Galois in 1832.) This exercise is referred to in Chapter 7 .

Varsha Aggarwal

Numerade Educator

02:26

Problem 36

Let $\beta=(1,3,5,7,9,8,6)(2,4,10)$. What is the smallest positive integer $n$ for which $\beta^{n}=\beta^{-5} ?$

Nick Johnson

Numerade Educator

02:32

Problem 37

Let $\alpha=(1,3,5,7,9)(2,4,6)(8,10)$. If $\alpha^{m}$ is a 5 -cycle, what can you say about $m$ ?

Matthew Markham

Numerade Educator

00:59

Problem 38

Let $H=\left\{\beta \in S_{5} \mid \beta(1)=1\right.$ and $\left.\beta(3)=3\right\}$. Prove that $H$ is a subgroup of $S_{5}$. How many elements are in $H ?$ Is your argument valid when $S_{5}$ is replaced by $S_{n}$ for $n \geq 3$ ? How many elements are in $H$ when $S_{5}$, is replaced by $A_{n}$ for $n \geq 4$ ?

Varsha Aggarwal

Numerade Educator

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Problem 39

In $S_{4}$, find a cyclic subgroup of order 4 and a noncyclic subgroup of order 4 .

Nick Johnson

Numerade Educator

03:18

Problem 40

In $S_{3}$, find elements $\alpha$ and $\beta$ such that $|\alpha|=2,|\beta|=2$, and $|\alpha \beta|=3$.

Vivek Singh

Numerade Educator

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Problem 41

Find group elements $\alpha$ and $\beta$ in $S_{5}$ such that $|\alpha|=3,|\beta|=3$, and $|\alpha \beta|=5$

Victor Salazar

Numerade Educator

02:17

Problem 42

Represent the symmetry group of an equilateral triangle as a group of permutations of its vertices (see Example 3).

James Schroeder

Numerade Educator

03:25

Problem 43

Prove that $S_{n}$ is non-Abelian for all $n \geq 3$.

Julian Wong

Numerade Educator

02:22

Problem 44

Prove that $A_{n}$ is non-Abelian for all $n \geq 4$.

Julian Wong

Numerade Educator

01:31

Problem 45

For $n \geq 3$, let $H=\left\{\beta \in S_{n} \mid \beta(1)=1\right.$ or 2 and $\beta(2)=1$ or 2$\}$. Prove that $H$ is a subgroup of $S_{n}$. Determine $|H|$.

Varsha Aggarwal

Numerade Educator

01:27

Problem 46

Show that in $S_{7}$, the equation $x^{2}=(1234)$ has no solutions but the equation $x^{3}=(1234)$ has at least two.

Carson Merrill

Numerade Educator

01:49

Problem 47

If $(a b)$ and $(c d)$ are distinct 2 -cycles in $S_{n}$, prove that $(a b)$ and $(c d)$ commute if and only if they are disjoint.

William Mead

Numerade Educator

01:41

Problem 48

Let $\alpha$ and $\beta$ belong to $S_{n}$. Prove that $\beta \alpha \beta^{-1}$ and $\alpha$ are both even or both odd.

Adriano Chikande

Numerade Educator

00:27

Problem 49

Viewing the members of $D_{4}$ as a group of permutations of a square labeled $1,2,3,4$ as described in Example 3, which geometric symmetries correspond to even permutations?

Sheryl Ezze

Numerade Educator

01:00

Problem 50

Viewing the members of $D_{5}$ as a group of permutations of a regular pentagon with consecutive vertices labeled $1,2,3,4,5$, what geometric symmetry corresponds to the permutation (14253)? Which symmetry corresponds to the permutation ( 25$)(34)$ ?

Victor Salazar

Numerade Educator

03:06

Problem 51

Let $n$ be an odd integer greater than 1 . Viewing $D_{n}$ as a group of permutations of a regular $n$ -gon with consecutive vertices labeled $1,2, \ldots, n$, explain why the rotation subgroup of $D_{n}$ is a subgroup of $A_{n}$

Sriparna Bhattacharjee

Numerade Educator

01:42

Problem 52

Let $\alpha_{1}, \alpha_{2}$ and $\alpha_{3}$ be 2 -cycles. Prove that $\alpha_{1} \alpha_{2} \alpha_{3} \neq \epsilon$. Generalize.

Urvashi Arora

Numerade Educator

03:01

Problem 53

Show that $A_{5}$ has 24 elements of order 5,20 elements of order 3 , and 15 elements of order $2 .$ (This exercise is referred to in Chapter 25.)

Dale Sanford

Numerade Educator

00:14

Problem 54

Find a cyclic subgroup of $A_{8}$ that has order 4 . Find a noncyclic subgroup of $A_{8}$ that has order 4

Isaac Chiu

Numerade Educator

03:06

Problem 55

Show that a permutation with odd order must be an even permutation.

Sriparna Bhattacharjee

Numerade Educator

00:53

Problem 56

Compute the order of each member of $A_{4}$. What arithmetic relationship do these orders have with the order of $A_{4} ?$

Prashansha Kaushik

Numerade Educator

08:30

Problem 57

Show that every element in $A_{n}$ for $n \geq 3$ can be expressed as a 3-cycle or a product of 3 -cycles.

Bobby Barnes

University of North Texas

02:19

Problem 58

Show that for $n \geq 3, Z\left(S_{n}\right)=\{\varepsilon\}$.

Zachary Mitchell

Numerade Educator

09:26

Problem 59

Verify the statement made in the discussion of the Verhoeff check digit scheme based on $D_{5}$ that $a * \sigma(b) \neq b * \sigma(a)$ for distinct $a$ and $b$. Use this to prove that $\sigma^{i}(a) * \sigma^{i+1}(b) \neq \sigma^{i}(b) * \sigma^{i+1}(a)$ for all $i$. Prove that this implies that all transposition errors involving adjacent digits are detected.

Oswaldo Jiménez

Numerade Educator

02:31

Problem 60

Use the Verhoeff check-digit scheme based on $D_{5}$ to append a check digit to $45723 .$

Prathan Jarupoonphol

Numerade Educator

02:27

Problem 61

Prove that every element of $S_{n}(n>1)$ can be written as a product of elements of the form $(1 k)$.

Wendi Zhao

Numerade Educator

03:06

Problem 62

(Indiana College Mathematics Competition) A card-shuffling machine always rearranges cards in the same way relative to the order in which they were given to it. All of the hearts arranged in order from ace to king were put into the machine, and then the shuffled cards were put into the machine again to be shuffled. If the cards emerged in the order $10,9, \mathrm{Q}, 8, \mathrm{~K}, 3,4, \mathrm{~A}, 5, \mathrm{~J}, 6,2,7$, in what
order were the cards after the first shuffle?

Chai Santi

Numerade Educator

00:59

Problem 63

Determine integers $n$ for which $H=\left\{\alpha \in A_{n} \mid \alpha^{2}=\varepsilon\right\}$ is a subgroup of $A_{n}$

Varsha Aggarwal

Numerade Educator

08:20

Problem 64

Find five subgroups of $S_{5}$ of order 24 .

Ely Crowder

Numerade Educator

01:39

Problem 65

Why does the fact that the orders of the elements of $A_{4}$ are 1,2, and 3 imply that $\left|Z\left(A_{4}\right)\right|=1 ?$

Christopher Stanley

Numerade Educator

01:01

Problem 66

Let $\alpha$ belong to $S_{n} .$ Prove that $|\alpha|$ divides $n !$

Nick Johnson

Numerade Educator

02:24

Problem 67

Encrypt the message ATTACK POSTPONED using the permutation $\alpha=\left[\begin{array}{lllll}1 & 2 & 3 & 4 & 5 \\ 2 & 1 & 5 & 3 & 4\end{array}\right]$

Bryan Lynn

Numerade Educator

02:24

Problem 68

The message VAADENWCNHREDEYA was encrypted using the permutation $\alpha=\left[\begin{array}{llll}1 & 2 & 3 & 4 \\ 2 & 4 & 1 & 3\end{array}\right]$. Decrypt it.

Bryan Lynn

Numerade Educator