Contemporary Abstract Algebra

Problem 2

Let $\alpha=\left[\begin{array}{llllllll}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ 2 & 3 & 4 & 5 & 1 & 7 & 8 & 6\end{array}\right]$ and $\beta=\left[\begin{array}{llllllll}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ 1 & 3 & 8 & 7 & 6 & 5 & 2 & 4\end{array}\right]$.
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

01:12

Problem 8

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

Tanishq Gupta

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

00:26

Problem 10

What is the maximum order of any element in $A_{10} ?$

Catherine Lemar

Numerade Educator

01:24

Problem 11

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

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

02:04

Problem 14

Find eight elements in $S_{6}$ that commute with (12) (34)(56). Do they form a subgroup of $S_{6}$ ?

David Collins

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

01:24

Problem 21

Let $\sigma$ be the permutation of the letters A through $Z$ that takes each letter to the one directly below it in the display following. Write $\sigma$ in cycle form.
AB C DE F G H I J KL M N O P QR S T U V W X Y Z
H D B G J E C M I L O N P F K R U S A W Q T V Z X Y

Vishnu P

Numerade Educator

00:26

Problem 22

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

Sam Limsuwannarot

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

Problem 24

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

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

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

09:50

Problem 27

Use Table $5.1$ to compute the following.
a. The centralizer of $\alpha_{3}=(13)(24)$
b. The centralizer of $\alpha_{12}=(124)$

Gaurav Kalra

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.

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

Problem 32

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

Victor Salazar

Numerade Educator

01:25

Problem 33

Find three elements $\sigma$ in $S_{9}$ with the property that $\sigma^{3}=$ $(157)(283)(469) .$

Aman Gupta

Numerade Educator

02:48

Problem 34

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

Gregory Higby

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 $\operatorname{stab}(a)$ 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

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

01:25

Problem 39

How many elements of order 5 are there in $A_{6} ?$

Gregory Higby

Numerade Educator

Problem 40

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

Nick Johnson

Numerade Educator

Problem 41

Suppose that $\beta$ is a 10 -cycle. For which integers $i$ between 2 and 10 is $\beta^{i}$ also a 10 -cycle?

Victor Salazar

Numerade Educator

03:18

Problem 42

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

Vivek Singh

Numerade Educator

Problem 43

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 44

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 45

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

Julian Wong

Numerade Educator

02:22

Problem 46

Prove that $A$ is non-Abelian for all $n \geq 4$.

Julian Wong

Numerade Educator

01:31

Problem 47

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 48

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 49

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

02:41

Problem 50

Let $\alpha$ be a 2 -cycle and $\beta$ be a $t$ -cycle in $S_{n}$. Prove that $\alpha \beta \alpha$ is a $t$ -cycle.

Tanishq Gupta

Numerade Educator

07:13

Problem 51

Use the previous exercise to prove that, if $\alpha$ and $\beta$ belong to $S_{n}$ and $\beta$ is the product of $k$ -cycles of lengths $n_{1}, n_{2}, \ldots, n_{k}$, then $\alpha \beta \alpha^{-1}$ is the product of $k$ -cycles of lengths $n_{1}, n_{2}, \ldots n_{k}$

Urvashi Arora

Numerade Educator

01:41

Problem 52

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

02:21

Problem 53

What is the smallest positive integer $n$ such that $S_{n}$ has an element of order greater than $2 n$ ?

Saurabh Chandra

Numerade Educator

02:55

Problem 54

Let $n$ be an even positive integer. Prove that $A_{n}$ has an element of order greater than $n$ if and only if $n \geq 8$.

James Chok

Numerade Educator

02:20

Problem 55

Let $n$ be an odd positive integer. Prove that $A_{n}$ has an element of order greater than $2 n$ if and only if $n \geq 13$.

James Chok

Numerade Educator

03:26

Problem 56

Let $n$ be an even positive integer. Prove that $A_{n}$ has an element of order greater than $2 n$ if and only if $n \geq 14$.

Clayton Schubring

Numerade Educator

00:27

Problem 57

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 58

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

Problem 59

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

03:30

Problem 60

Let $n$ be an integer greater than $1 .$ Viewing $D_{n}$ as a group of permutations of a regular $n$ -gon with consecutive vertices labeled $1,2, \ldots$, $n$, determine for which $n$ all the permutations corresponding to reflections in $D_{n}$ are even permutations. Hint: Consider the fours cases for $n$ mod 4 .

Wen Zheng

Numerade Educator

03:01

Problem 61

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 62

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

Isaac Chiu

Numerade Educator

Problem 63

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

Nick Johnson

Numerade Educator

00:53

Problem 64

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

05:55

Problem 65

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

Bobby Barnes

University of North Texas

02:19

Problem 66

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

Zachary Mitchell

Numerade Educator

09:26

Problem 67

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 68

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

Prathan Jarupoonphol

Numerade Educator

02:27

Problem 69

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

Problem 70

(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

Problem 71

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

Sriparna Bhattacharjee

Numerade Educator

01:07

Problem 72

Let $G$ be a group. Prove or disprove that $H=\left\{g^{2} \mid g \in G\right\}$ is a subgroup of $G$. (Compare with Example 5 in Chapter 3.)

Varsha Aggarwal

Numerade Educator

01:18

Problem 73

Let $H=\left\{\alpha^{2} \mid \alpha \in S_{4}\right\}$ and $K=\left\{\alpha^{2} \mid \alpha \in S_{5}\right\}$. Prove $H=A_{4}$ and
$K=A_{5}$.

Tyler Moulton

Numerade Educator

02:20

Problem 74

Let $H=\left\{\alpha^{2} \mid \alpha \in S_{6}\right\}$. Prove $H \neq A_{6}$.

Sandip Ranjan

Numerade Educator

Problem 75

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

06:12

Problem 76

Given that $\beta$ and $\gamma$ are in $S_{4}$ with $\beta \gamma=(1432), \gamma \beta=(1243)$, and $\beta(1)=4$, determine $\beta$ and $\gamma$.

Ruby P

Numerade Educator

01:39

Problem 77

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

08:20

Problem 78

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

Ely Crowder

Numerade Educator

06:58

Problem 79

Find six subgroups of order 60 in $S_{6}$.

Dale Sanford

Numerade Educator

01:07

Problem 80

For $n>1$, let $H$ be the set of all permutations in $S_{n}$ that can be expressed as a product of a multiple of four transpositions. Show that $H=A_{n^{*}}$

Sneha Ravi

Numerade Educator

08:20

Problem 81

Shown below are four tire rotation patterns recommended by the Dunlop Tire Company. Explain how these patterns can be represented as permutations in $S_{4}$ and find the smallest subgroup of $S_{4}$ that contains these four patterns. Is the subgroup Abelian?

Ely Crowder

Numerade Educator

01:11

Problem 82

Label the four locations of tires on an automobile with the labels $1,2,3$, and 4, clockwise. Let $a$ represent the operation of switching the tires in positions 1 and 3 and switching the tires in positions 2 and 4 . Let $b$ represent the operation of rotating the tires in positions 2,3, and 4 clockwise and leaving the tire in position 1 as is. Let $G$ be the group of all possible combinations of $a$ and $b$. How many elements are in $G ?$

Heather Zimmers

Numerade Educator