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Contemporary Abstract Algebra

Joseph Gallian

Chapter 8

External Direct Products - all with Video Answers

Educators

Chapter Questions

01:35

Problem 1

Prove that the external direct product of any finite number of groups is a group. (This exercise is referred to in this chapter.)

Nick Johnson
Nick Johnson
Numerade Educator
02:45

Problem 2

Show that $Z_{2} \oplus Z_{2} \oplus Z_{2}$ has seven subgroups of order 2 .

Uma Kumari
Uma Kumari
Numerade Educator
01:07

Problem 3

Let $G$ be a group with identity $e_{G}$ and let $H$ be a group with identity $e_{H}$. Prove that $G$ is isomorphic to $G \oplus\left\{e_{H}\right\}$ and that $H$ is isomorphic to $\left\{e_{G}\right\} \oplus H$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:22

Problem 4

. Show that $G \oplus H$ is Abelian if and only if $G$ and $H$ are Abelian. State the general case.

Wendi Zhao
Wendi Zhao
Numerade Educator
02:45

Problem 5

Prove or disprove that $Z \oplus Z$ is a cyclic group.

Uma Kumari
Uma Kumari
Numerade Educator
02:45

Problem 6

Prove, by comparing orders of elements, that $Z_{8} \oplus Z_{2}$ is not isomorphic to $Z_{4} \oplus Z_{4}$.

Uma Kumari
Uma Kumari
Numerade Educator
View

Problem 7

Prove that $G_{1} \oplus G_{2}$ is isomorphic to $G_{2} \oplus G_{1} .$ State the general case.

Nick Johnson
Nick Johnson
Numerade Educator
03:54

Problem 8

Is $Z_{3} \oplus Z_{9}$ isomorphic to $Z_{27}$ ? Why?

Anthony Ramos
Anthony Ramos
Numerade Educator
03:54

Problem 9

Is $Z_{3} \oplus Z_{5}$ isomorphic to $Z_{15}$ ? Why?

Anthony Ramos
Anthony Ramos
Numerade Educator
03:13

Problem 10

How many elements of order 9 does $Z_{3} \oplus Z_{9}$ have? (Do not do this exercise by brute force.)

Megan Mcfarland
Megan Mcfarland
Numerade Educator
00:40

Problem 11

How many elements of order 4 does $Z_{4} \oplus Z_{4}$ have? (Do not do this by examining each element.) Explain why $Z_{4} \oplus Z_{4}$ has the same number of elements of order 4 as does $Z_{8000000} \oplus Z_{400000^{-}}$ Generalize to the case $Z_{m} \oplus Z_{n^{\prime}}$

Hunza Gilgit
Hunza Gilgit
Numerade Educator
06:02

Problem 12

Give examples of four groups of order 12, no two of which are isomorphic. Give reasons why no two are isomorphic.

Ely Crowder
Ely Crowder
Numerade Educator
03:35

Problem 13

For each integer $n>1$, give examples of two nonisomorphic groups of order $n^{2}$

Gaurav Kalra
Gaurav Kalra
Numerade Educator
08:25

Problem 14

The dihedral group $D_{n}$ of order $2 n(n \geq 3)$ has a subgroup of $n$ rotations and a subgroup of order $2 .$ Explain why $D_{n}$ cannot be isomorphic to the external direct product of two such groups.

Ely Crowder
Ely Crowder
Numerade Educator
00:49

Problem 15

Prove that the group of complex numbers under addition is isomorphic to $\mathbf{R} \oplus \mathbf{R}$.

Amy Jiang
Amy Jiang
Numerade Educator
02:37

Problem 16

Suppose that $G_{1} \approx G_{2}$ and $H_{1} \approx H_{2} .$ Prove that $G_{1} \oplus H_{1} \approx G_{2} \oplus$ $H_{2}$. State the general case.

Akash Goyal
Akash Goyal
Numerade Educator
01:22

Problem 17

If $G \oplus H$ is cyclic, prove that $G$ and $H$ are cyclic. State the general case.

Wendi Zhao
Wendi Zhao
Numerade Educator
03:10

Problem 18

In $Z_{40} \oplus Z_{30}$, find two subgroups of order $12 .$

Gaurav Kalra
Gaurav Kalra
Numerade Educator
01:40

Problem 19

If $r$ is a divisor of $m$ and $s$ is a divisor of $n$, find a subgroup of $Z_{m} \oplus$ $Z_{n}$ that is isomorphic to $Z_{r} \oplus Z_{s}$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:46

Problem 20

Find a subgroup of $Z_{12} \oplus Z_{18}$ that is isomorphic to $Z_{9} \oplus Z_{4}$.

Yuva S
Yuva S
Numerade Educator
01:14

Problem 21

let $G$ and $H$ be finite groups and $(g, h) \in G \oplus H$. State a necessary and sufficient condition for $\langle(g, h)\rangle=\langle g\rangle \oplus\langle h\rangle$.

Mohamed Mohamed
Mohamed Mohamed
Numerade Educator
02:41

Problem 22

Determine the number of elements of order 15 and the number of cyclic subgroups of order 15 in $Z_{30} \oplus Z_{20}$.

Babita Kumari
Babita Kumari
Numerade Educator
06:15

Problem 23

What is the order of any nonidentity element of $Z_{3} \oplus Z_{3} \oplus Z_{3}$ ? Generalize.

Uma Kumari
Uma Kumari
Numerade Educator
00:46

Problem 24

Let $m>2$ be an even integer and let $n>2$ be an odd integer. Find a formula for the number of elements of order 2 in $D_{m} \oplus D_{n^{\circ}}$

Jay Patel
Jay Patel
Numerade Educator
08:25

Problem 25

Let $M$ be the group of all real $2 \times 2$ matrices under addition. Let $N=\mathbf{R} \oplus \mathbf{R} \oplus \mathbf{R} \oplus \mathbf{R}$ under componentwise addition. Prove that $M$ and $N$ are isomorphic. What is the corresponding theorem for the group of $m \times n$ matrices under addition?

Ely Crowder
Ely Crowder
Numerade Educator
02:28

Problem 26

The group $S_{3} \oplus Z_{2}$ is isomorphic to one of the following groups:
$Z_{12}, Z_{6} \oplus Z_{2}, A_{4}, D_{6}^{-}$. Determine which one by elimination.

Brandon Collins
Brandon Collins
Numerade Educator
01:51

Problem 27

Let $G$ be a group, and let $H=\{(g, g) \mid g \in G\}$. Show that $H$ is a subgroup of $G \oplus G$. (This subgroup is called the diagonal of $G \oplus G .$ When $G$ is the set of real numbers under addition, describe $G \oplus G$ and $H$ geometrically.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:07

Problem 28

Find a subgroup of $Z_{4} \oplus Z_{2}$ that is not of the form $H \oplus K$, where $H$ is a subgroup of $Z_{4}$ and $K$ is a subgroup of $Z_{2}$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
06:15

Problem 29

Find all subgroups of order 3 in $Z_{9} \oplus Z_{3}$.

Uma Kumari
Uma Kumari
Numerade Educator
00:40

Problem 30

Find all subgroups of order 4 in $Z_{A} \oplus Z_{A}$.

Hunza Gilgit
Hunza Gilgit
Numerade Educator
02:10

Problem 31

What is the largest order of any element in $Z_{30} \oplus Z_{20}$ ?

Kim Trang Nguyen
Kim Trang Nguyen
Numerade Educator
04:40

Problem 32

What is the order of the largest cyclic subgroup of $Z_{6} \oplus Z_{10} \oplus Z_{15}$ ? What is the order of the largest cyclic subgroup of $Z_{n_{1}} \oplus Z_{n_{2}} \oplus \cdots$ $\oplus Z_{n_{k}} ?$

Uma Kumari
Uma Kumari
Numerade Educator
06:15

Problem 33

Find three cyclic subgroups of maximum possible order in $Z_{6} \oplus$ $Z_{10} \oplus Z_{15}$ of the form $\langle a\rangle \oplus\langle b\rangle \oplus\langle c\rangle$, where $a \in Z_{6}, b \in Z_{10}$, and
$c \in Z_{15^{\circ}}$

Uma Kumari
Uma Kumari
Numerade Educator
04:30

Problem 34

How many elements of order 2 are in $Z_{2000000} \oplus Z_{4000000}$ ? Generalize.

Uma Kumari
Uma Kumari
Numerade Educator
01:46

Problem 35

Find a subgroup of $Z_{800} \oplus Z_{200}$ that is isomorphic to $Z_{2} \oplus Z_{4}$.

Yuva S
Yuva S
Numerade Educator
04:30

Problem 36

Find a subgroup of $Z_{12} \oplus Z_{4} \oplus Z_{15}$ that has order 9 .

Uma Kumari
Uma Kumari
Numerade Educator
02:24

Problem 37

Prove that $\mathbf{R}^{*} \oplus \mathbf{R}^{*}$ is not isomorphic to $\mathbf{C}^{*}$. (Compare this with Exercise 15.)

Aman Gupta
Aman Gupta
Numerade Educator
06:19

Problem 38

Let
$$H=\left\{\left[\begin{array}{lll}1 & a & b \\0 & 1 & 0 \\
0 & 0 & 1\end{array}\right] \mid a, b \in Z_{3}\right\}$$

Fan Yang
Fan Yang
Numerade Educator
View

Problem 39

Let $G=\left\{3^{m} 6^{n} \mid m, n \in Z\right\}$ under multiplication. Prove that $G$ is isomorphic to $Z \oplus Z$. Does your proof remain valid if $G=\left\{3^{m} 9^{n} \mid m, n \in Z\right\} ?$

Nick Johnson
Nick Johnson
Numerade Educator
02:54

Problem 40

Let $\left(a_{1}, a_{2}, \ldots, a_{n}\right) \in G_{1} \oplus G_{2} \oplus \cdots \oplus G_{n^{*}}$ Give a necessary and
sufficient condition for $\left|\left(a_{1}, a_{2}, \ldots, a_{n}\right)\right|=\infty$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:26

Problem 41

Prove that $D_{3} \oplus D_{4} \neq D_{12} \oplus Z_{2}$.

Erika Bustos
Erika Bustos
Numerade Educator
01:29

Problem 42

Determine the number of cyclic subgroups of order 15 in $Z_{90} \oplus Z_{36}$. Provide a generator for each of the subgroups of order 15 .

Ashley Volpe
Ashley Volpe
Numerade Educator
04:51

Problem 43

List the elements in the groups $U_{5}(35)$ and $U_{7}(35)$.

Dr. Satish Ingale
Dr. Satish Ingale
Numerade Educator
04:51

Problem 44

List the elements in the groups $U_{5}(35)$ and $U_{7}(35)$.

Dr. Satish Ingale
Dr. Satish Ingale
Numerade Educator
01:58

Problem 45

Prove or disprove that $\boldsymbol{C}^{*}$ has a subgroup isomorphic to $Z_{2} \oplus Z_{2}$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:40

Problem 46

Let $G$ be a group isomorphic to $Z_{n_{1}} \oplus Z_{n_{2}} \oplus \ldots \oplus Z_{n k}$. Let $x$ be the product of all elements in $G$. Describe all possibilities for $x$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:25

Problem 47

If a group has exactly 24 elements of order 6 , how many cyclic subgroups of order 6 does it have?

Gregory Higby
Gregory Higby
Numerade Educator
01:40

Problem 48

For any Abelian group $G$ and any positive integer $n$, let $G^{n}=\left\{g^{n} \mid\right.$ $g \in G\}$ (see Exercise 17, Supplementary Exercises for Chapters $1-4$ ). If $H$ and $K$ are Abelian, show that $(H \oplus K)^{n}=H^{n} \oplus K^{n}$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:30

Problem 49

Express $\operatorname{Aut}(U(25))$ in the form $Z_{m} \oplus Z_{n}$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
04:30

Problem 50

Determine $\operatorname{Aut}\left(Z, \oplus Z_{2}\right)$.

Uma Kumari
Uma Kumari
Numerade Educator
01:27

Problem 51

. Suppose that $n_{1}, n_{2}, \ldots, n_{k}$ are positive even integers. How many elements of order 2 does $Z_{n_{1}} \oplus Z_{n_{2}} \oplus \cdots \oplus Z_{n_{k}}$ have ? How many are there if we drop the requirement that $n_{1}, n_{2}, \ldots, n_{k}$ must be even?

Anthony Ramos
Anthony Ramos
Numerade Educator
04:30

Problem 52

Is $Z_{10} \oplus Z_{12} \oplus Z_{6} \approx Z_{60} \oplus Z_{6} \oplus Z_{2} ?$

Uma Kumari
Uma Kumari
Numerade Educator
04:30

Problem 53

Is $Z_{10} \oplus Z_{12} \oplus Z_{6} \approx Z_{15} \oplus Z_{4} \oplus Z_{12} ?$

Uma Kumari
Uma Kumari
Numerade Educator
01:05

Problem 54

Find an isomorphism from $Z_{12}$ to $Z_{4} \oplus Z_{3}$.

ES
Ellie Sun
Numerade Educator
01:05

Problem 55

How many isomorphisms are there from $Z_{12}$ to $Z_{4} \oplus Z_{3}$ ?

ES
Ellie Sun
Numerade Educator
06:15

Problem 56

Suppose that $\phi$ is an isomorphism from $Z_{3} \oplus Z_{5}$ to $Z_{15}$ and $\phi(2,3)=2$. Find the element in $Z_{3} \oplus Z_{5}$ that maps to 1 .

Uma Kumari
Uma Kumari
Numerade Educator
01:59

Problem 57

If $\phi$ is an isomorphism from $Z_{4} \oplus Z_{3}$ to $Z_{12}$, what is $\phi(2,0) ?$ What are the possibilities for $\phi(1,0) ?$ Give reasons for your answer.

Sherrie Fenner
Sherrie Fenner
Numerade Educator
01:54

Problem 58

Prove that $Z_{5} \oplus Z_{5}$ has exactly six subgroups of order 5 .

Raushan Kumar
Raushan Kumar
Numerade Educator
07:18

Problem 59

Let $(a, b)$ belong to $Z_{m} \oplus Z_{n} .$ Prove that $1(a, b) \mid$ divides $\operatorname{lcm}(m, n)$.

Bryan Lynn
Bryan Lynn
Numerade Educator
02:55

Problem 60

. Let $G=\left\{a x^{2}+b x+c \mid a, b, c \in Z_{3}\right\} .$ Add elements of $G$ as you would polynomials with integer coefficients, except use modulo 3 addition. Prove that $G$ is isomorphic to $Z_{3} \oplus Z_{3} \oplus Z_{3}$. Generalize.

Nick Johnson
Nick Johnson
Numerade Educator
06:41

Problem 61

Determine all cyclic groups that have exactly two generators.

Ely Crowder
Ely Crowder
Numerade Educator
00:26

Problem 62

Explain a way that a string of length $n$ of the four nitrogen bases A, $\mathrm{T}, \mathrm{G}$, and $\mathrm{C}$ could be modeled with the external direct product of $n$ copies of $Z_{2} \oplus Z_{2}$

Eleanor Behling
Eleanor Behling
Numerade Educator
06:02

Problem 63

Let $p$ be a prime. Prove that $Z_{p} \oplus Z_{p}$ has exactly $p+1$ subgroups of order $p$.

Ely Crowder
Ely Crowder
Numerade Educator
06:02

Problem 64

Give an example of an infinite non-Abelian group that has exactly six elements of finite order.

Ely Crowder
Ely Crowder
Numerade Educator
03:09

Problem 65

Give an example to show that there exists a group with elements $a$ and $b$ such that $|a|=\infty,|b|=\infty$, and $|a b|=2$.

Angelo Rendina
Angelo Rendina
Numerade Educator
01:02

Problem 66

Express $U(165)$ as an external direct product of cyclic groups of the form $Z_{n}$.

Amy Jiang
Amy Jiang
Numerade Educator
01:56

Problem 67

Express $U(165)$ as an external direct product of $U$ -groups in four different ways.

Jeyasree R T
Jeyasree R T
Numerade Educator
01:57

Problem 68

Without doing any calculations in $\operatorname{Aut}\left(Z_{20}\right)$, determine how many elements of $\operatorname{Aut}\left(Z_{20}\right)$ have order $4 .$ How many have order $2 ?$

Sherrie Fenner
Sherrie Fenner
Numerade Educator
01:14

Problem 69

Without doing any calculations in $\operatorname{Aut}\left(Z_{720}\right)$, determine how many elements of $\operatorname{Aut}\left(Z_{720}\right)$ have order 6 .

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
00:33

Problem 70

Without doing any calculations in $U(27)$, decide how many subgroups $U(27)$ has.

K B
K B
Numerade Educator
02:29

Problem 71

What is the largest order of any element in $U(900)$ ?

Emily Harris
Emily Harris
Numerade Educator
03:56

Problem 72

Let $p$ and $q$ be odd primes and let $m$ and $n$ be positive integers. Explain why $U\left(p^{m}\right) \oplus U\left(q^{n}\right)$ is not cyclic.

Sriparna Bhattacharjee
Sriparna Bhattacharjee
Numerade Educator
03:58

Problem 73

Use the results presented in this chapter to prove that $U(55)$ is isomorphic to $U(75)$.

Anthony Ramos
Anthony Ramos
Numerade Educator
03:58

Problem 74

Use the results presented in this chapter to prove that $U(144)$ is isomorphic to $U(140)$.

Anthony Ramos
Anthony Ramos
Numerade Educator
01:58

Problem 75

For every $n>2$, prove that $U(n)^{2}=\left\{x^{2} \mid x \in U(n)\right\}$ is a proper subgroup of $U(n)$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
00:39

Problem 76

Show that $U(55)^{3}=\left\{x^{3} \mid x \in U(55)\right\}$ is $U(55)$.

Rujula Deshmukh
Rujula Deshmukh
Numerade Educator
03:30

Problem 77

Find an integer $n$ such that $U(n)$ contains a subgroup isomorphic to $Z_{5} \oplus Z_{5}$

Julian Wong
Julian Wong
Numerade Educator
01:40

Problem 78

Find a subgroup of order 6 in $U(700)$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:45

Problem 79

Show that there is a $U$ -group containing a subgroup isomorphic to $Z_{3} \oplus Z_{3 \cdot}$

Uma Kumari
Uma Kumari
Numerade Educator
01:46

Problem 80

Find an integer $n$ such that $U(n)$ is isomorphic to $Z_{2} \oplus Z_{4} \oplus Z_{9}$.

Yuva S
Yuva S
Numerade Educator
01:26

Problem 81

What is the smallest positive integer $k$ such that $x^{k}=e$ for all $x$ in $U(7 \cdot 17) ?$ Generalize to $U(p q)$ where $p$ and $q$ are distinct primes.

Nick Johnson
Nick Johnson
Numerade Educator
01:15

Problem 82

If $k$ divides $m$ and $m$ divides $n$, how are $U_{m}(n)$ and $U_{k}(n)$ related?

Nishant Tyagi
Nishant Tyagi
Numerade Educator
02:22

Problem 83

Let $p_{1}, p_{2}, \ldots, p_{k}$ be distinct odd primes and $n_{1}, n_{2}, \ldots, n_{k}$ be positive integers. Determine the number of elements of order 2 in $U\left(p_{1}^{n_{1}} p_{2}^{n_{2}} \cdots p_{k}^{n_{k}}\right)$. How many are there in $U\left(2^{n} p_{1}^{n_{1}} p_{2}^{n_{2}} \cdots p_{k}^{n_{k}}\right)$ where
$n$ is at least 3 ?

James Chok
James Chok
Numerade Educator
01:02

Problem 84

Show that no $U$ -group has order 14 .

Adriano Chikande
Adriano Chikande
Numerade Educator
08:50

Problem 85

Show that there is a $U$ -group containing a subgroup isomorphic to $Z_{14}$

Ely Crowder
Ely Crowder
Numerade Educator
01:46

Problem 86

Show that no $U$ -group is isomorphic to $Z_{4} \oplus Z_{4}$.

Yuva S
Yuva S
Numerade Educator
01:40

Problem 87

. Show that there is a $U$ -group containing a subgroup isomorphic to $Z_{4} \oplus Z_{4}$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
09:11

Problem 88

Using the RSA scheme with $p=37, q=73$, and $e=5$, what number would be sent for the message "RM"?

Bryan Lynn
Bryan Lynn
Numerade Educator
09:11

Problem 89

Assuming that a message has been sent via the RSA scheme with $p=37, q=73$, and $e=5$, decode the received message "34."

Bryan Lynn
Bryan Lynn
Numerade Educator