• Home
  • Textbooks
  • Contemporary Abstract Algebra
  • Normal Subgroups and Factor Groups

Contemporary Abstract Algebra

Joseph Gallian

Chapter 9

Normal Subgroups and Factor Groups - all with Video Answers

Educators

Chapter Questions

04:01

Problem 1

Let $H=\{(1),(12)\} .$ Is $H$ normal in $S_{3} ?$

Ethan Somes
Ethan Somes
Numerade Educator
00:47

Problem 2

Prove that $A_{n}$ is normal in $S_{n}$.

Linh Vu
Linh Vu
Numerade Educator
01:11

Problem 3

In $D_{4}$, let $K=\left\{R_{0}, R_{90}, R_{180}, R_{270}\right\} .$ Write $H R_{90}$ in the form $x H$, where $x \in K$. Write $D R_{270}$ in the form $x D$, where $x \in K .$ Write $R_{90} V$ in the form $V x$, where $x \in K$

Babita Kumari
Babita Kumari
Numerade Educator
00:32

Problem 4

Write $(12)(13)(14)$ in the form $\alpha(12)$, where $\alpha \in A_{4} .$ Write $(1234)$ (12) (23), in the form $\alpha(1234)$, where $\alpha \in A_{4}$.

Sriparna Bhattacharjee
Sriparna Bhattacharjee
Numerade Educator
08:37

Problem 5

Show that if $G$ is the internal direct product of $H_{1}, H_{2}, \ldots, H_{n}$ and $i \neq j$ with $1 \leq i \leq n, 1 \leq j \leq n$, then $H_{i} \cap H_{j}=\{e\} .$ (This exercise is referred to in this chapter.)

Ahmad Reda
Ahmad Reda
Numerade Educator
01:42

Problem 6

Let $H=\left\{\left[\begin{array}{ll}a & b \\ 0 & d\end{array}\right] \mid a, b, d \in \mathbf{R}, a d \neq 0\right\} .$ Is $H$ a normal sub-
group of $G L(2, \mathbf{R}) ?$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:07

Problem 7

Let $G=G L(2, \mathbf{R})$ and let $K$ be a subgroup of $\mathbf{R}^{*}$. Prove that $H=$ $\{A \in G \mid \operatorname{det} A \in K\}$ is a normal subgroup of $G$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
05:10

Problem 8

Viewing $\langle 3\rangle$ and $\langle 12\rangle$ as subgroups of $Z$, prove that $\langle 3\rangle /\langle 12\rangle$ is isomorphic to $Z_{4} .$ Similarly, prove that $\langle 8\rangle /\langle 48\rangle$ is isomorphic to $Z_{6}$. Generalize to arbitrary integers $k$ and $n .$

Ely Crowder
Ely Crowder
Numerade Educator
04:03

Problem 9

Prove that if $H$ has index 2 in $G$, then $H$ is normal in $G$. (This exercise is referred to in Chapters 24 and 25 and this chapter.)

Eric Mockensturm
Eric Mockensturm
Numerade Educator
01:02

Problem 10

Let $H=\{(1),(12)(34)\}$ in $A_{4}$.
a. Show that $H$ is not normal in $A_{4}$.
b. Referring to the multiplication table for $A_{4}$ in Table $5.1$ on page 105, show that, although $\alpha_{6} H=\alpha_{7} H$ and $\alpha_{9} H=\alpha_{11} H$, it is not true that $\alpha_{6} \alpha_{9} H=\alpha_{7} \alpha_{11} H .$ Explain why this proves that the left cosets of $H$ do not form a group under coset multiplication.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:35

Problem 11

Prove that a factor group of a cyclic group is cyclic.

Nick Johnson
Nick Johnson
Numerade Educator
View

Problem 12

Prove that a factor group of an Abelian group is Abelian.

Nick Johnson
Nick Johnson
Numerade Educator
01:40

Problem 13

Let $H$ be a normal subgroup of a finite group $G$ and let $a$ be an element of $G$. Complete the following statement: The order of the element $a H$ in the factor group $G / H$ is the smallest positive integer $n$ such that $a^{n}$ is _____.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
00:48

Problem 14

What is the order of the element $14+\langle 8\rangle$ in the factor group $Z_{24} /\langle 8\rangle ?$

Erika Bustos
Erika Bustos
Numerade Educator
01:00

Problem 15

What is the order of the element $4 U_{5}(105)$ in the factor group $U(105) / U_{5}(105) ?$

Pronoy Sinha
Pronoy Sinha
Numerade Educator
01:25

Problem 16

Recall that $Z\left(D_{6}\right)=\left\{R_{0}, R_{180}\right\} .$ What is the order of the element $R_{60} Z\left(D_{6}\right)$ in the factor group $D_{6} / Z\left(D_{6}\right)$ ?

Chelsea Hoke
Chelsea Hoke
Numerade Educator
01:37

Problem 17

Let $G=Z /\langle 20\rangle$ and $H=\langle 4\rangle /\langle 20\rangle$. List the elements of $H$ and $G / H$.

James Kiss
James Kiss
Numerade Educator
01:21

Problem 18

What is the order of the factor group $Z_{60} /\langle 15\rangle ?$

Nez Nikoo
Nez Nikoo
Numerade Educator
00:46

Problem 19

Determine all normal subgroups of $D_{n}$ of order $2 .$

Nick Johnson
Nick Johnson
Numerade Educator
00:53

Problem 20

List the elements of $U(20) / U_{5}(20)$.

Sri Datta Vikas Buchemmavari
Sri Datta Vikas Buchemmavari
Numerade Educator
06:02

Problem 21

Prove that an Abelian group of order 33 is cyclic. Does your proof hold when 33 is replaced by $p q$ where $p$ and $q$ are distinct primes?

Ely Crowder
Ely Crowder
Numerade Educator
01:06

Problem 22

Determine the order of $(Z \oplus Z) /\langle(2,2)\rangle$. Is the group cyclic?

Hunza Gilgit
Hunza Gilgit
Numerade Educator
01:07

Problem 23

Let $G_{1}$ and $G_{2}$ be finite groups. If $H_{1}$ is a normal subgroup of $G_{1}$ and $H_{2}$ is a normal subgroup of $G_{2}$ give a formula for $\left|G_{1} / H_{1} \oplus G_{2} / H_{2}\right|$ in terms of $\left|G_{1}\right|,\left|G_{2}\right|,\left|H_{1}\right|$ and $\left|H_{2}\right|$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:28

Problem 24

The group $\left(Z_{4} \oplus Z_{12}^{2}\right) /\langle(2,2)\rangle$ is isomorphic to one of $Z_{8}, Z_{4} \oplus Z_{2}$, or $Z_{2} \oplus Z_{2} \oplus Z_{2}$. Determine which one by elimination.

Brandon Collins
Brandon Collins
Numerade Educator
02:28

Problem 25

Let $G=U(32)$ and $H=\{1,15\}$. The group $G / H$ is isomorphic to one of $Z_{8}, Z_{4} \oplus Z_{2}$, or $Z_{2} \oplus Z_{2} \oplus Z_{2} .$ Determine which one by elimination.

Brandon Collins
Brandon Collins
Numerade Educator
01:40

Problem 26

Let $H=\{1,17,41,49,73,89,97,113\}$ under multiplication modulo 120 . Write $H$ as a external direct product of groups of the form $Z_{2} k .$ Write $H$ as an internal direct product of nontrivial subgroups.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
03:58

Problem 27

Let $G=U(16), H=\{1,15\}$, and $K=\{1,9\} .$ Are $H$ and $K$ isomorphic? Are $G / H$ and $G / K$ isomorphic?

Anthony Ramos
Anthony Ramos
Numerade Educator
02:28

Problem 28

Let $G=Z_{4} \oplus Z_{4}, H=\{(0,0),(2,0),(0,2),(2,2)\}$, and $K=\langle(1,2)\rangle$. Is $G / H$ isomorphic to $Z_{4}$ or $Z_{2} \oplus Z_{2} ?$ Is $G / K$ isomorphic to $Z_{4}$ or $Z_{2} \oplus Z_{2} ?$

Brandon Collins
Brandon Collins
Numerade Educator
02:05

Problem 29

Explain why a non-Abelian group of order 8 cannot be the internal direct product of proper subgroups.

Wendi Zhao
Wendi Zhao
Numerade Educator
01:56

Problem 30

Express $U(165)$ as an internal direct product of proper subgroups in four different ways.

Jeyasree R T
Jeyasree R T
Numerade Educator
02:55

Problem 31

Let $\mathbf{R}^{*}$ denote the group of all nonzero real numbers under multiplication. Let $\mathbf{R}^{+}$ denote the group of positive real numbers under multiplication. Prove that $\mathbf{R}^{*}$ is the internal direct product of $\mathbf{R}^{+}$ and the subgroup $\{1,-1\}$.

Nick Johnson
Nick Johnson
Numerade Educator
00:59

Problem 32

If $N$ is a normal subgroup of $G$ and $|G / N|=m$, show that $x^{m} \in N$ for all $x$ in $G$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:05

Problem 33

Let $H$ and $K$ be subgroups of a group $G$. If $G=H K$ and $g=h k$, where $h \in H$ and $k \in K$, is there any relationship among $|g|,|h|$ and $|k| ?$ What if $G=H \times K$ ?

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
07:57

Problem 34

In $Z$, let $H=\langle 5\rangle$ and $K=\langle 7\rangle$. Prove that $Z=H K$. Does $Z=H \times K$ ?

Sandip Ranjan
Sandip Ranjan
Numerade Educator
05:39

Problem 35

Let $G=\left\{3^{a} 6^{b} 10^{c} \mid a, b, c \in Z\right\}$ under multiplication and $H=$ $\left\{3^{a} 6^{b} 12^{c} \mid a, b, c \in Z\right\}$ under multiplication. Prove that $G=\langle 3\rangle \times$ $\langle 6\rangle \times\langle 10\rangle$, whereas $H \neq\langle 3\rangle \times\langle 6\rangle \times\langle 12\rangle$

NW
Nida Wasiq
Numerade Educator
01:58

Problem 36

Determine all subgroups of $\mathbf{R}^{*}$ (nonzero reals under multiplication) of index 2 .

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:07

Problem 37

Let $G$ be a finite group and let $H$ be a normal subgroup of $G$. Prove that the order of the element $g H$ in $G / H$ must divide the order of $g$ in $G$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:40

Problem 38

Prove that for every positive integer $n, Q / Z$ has an element of order $n .$

Adriano Chikande
Adriano Chikande
Numerade Educator
01:02

Problem 39

Let $H$ be a subgroup of a group $G$ with the property that for all $a$ and $b$ in $G, a H b H=a b H$. Prove that $H$ is a normal subgroup of $G$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:02

Problem 40

Let in $S_{3}$ let $H=\{(1),(12)\} .$ Show that $(13) H(23) H \neq(13)(23) H$. (This proves that when $H$ is not a normal subgroup of a group $G$, the product of two left cosets of $H$ in $G$ need not be a left coset of $H$ in $G$.)

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
03:40

Problem 41

Show that $Q$, the group of rational numbers under addition, has no proper subgroup of finite index.

Akash Goyal
Akash Goyal
Numerade Educator
01:51

Problem 42

An element is called a square if it can be expressed in the form $b^{2}$ for some $b$. Suppose that $G$ is an Abelian group and $H$ is a subgroup of $G$. If every element of $H$ is a square and every element of $G / H$ is a square, prove that every element of $G$ is a square. Does your proof remain valid when "square" is replaced by " $n$ th power" where $n$ is any integer?

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:02

Problem 43

Show, by example, that in a factor group $G / H$ it can happen that $a H=b H$ but $|a| \neq|b|$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:38

Problem 44

Verify that the mapping defined at the end of the proof of Theorem $9.6$ is an isomorphism.

Nick Johnson
Nick Johnson
Numerade Educator
06:02

Problem 45

Let $p$ be a prime. Show that if $H$ is a subgroup of a group of order $2 p$ that is not normal, then $H$ has order 2 .

Ely Crowder
Ely Crowder
Numerade Educator
01:05

Problem 46

Show that $D_{13}$ is isomorphic to $\operatorname{Inn}\left(\mathrm{D}_{13}\right)$.

Anthony Ramos
Anthony Ramos
Numerade Educator
02:05

Problem 47

Let $H$ and $K$ be subgroups of a group $G$. If $|H|=63$ and $|K|=45$, prove that $H \cap K$ is Abelian.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
05:17

Problem 48

If $G$ is a group and $|G: Z(G)|=4$, prove that $G / Z(G) \approx Z_{2} \oplus Z_{2}$.

Nick Johnson
Nick Johnson
Numerade Educator
06:47

Problem 49

Suppose that $G$ is a non-Abelian group of order $p^{3}$, where $p$ is a prime, and $Z(G) \neq\{e\}$. Prove that $|Z(G)|=p$.

Brandon Collins
Brandon Collins
Numerade Educator
01:13

Problem 50

If $|G|=p q$, where $p$ and $q$ are primes that are not necessarily distinct, prove that $|Z(G)|=1$ or $p q$.

WZ
Wen Zheng
Numerade Educator
01:07

Problem 51

Let $H$ be a normal subgroup of $G$ and $K$ a subgroup of $G$ that contains $H$. Prove that $K$ is normal in $G$ if and only if $K / H$ is normal in $G / H$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:02

Problem 52

Let $G$ be an Abelian group and let $H$ be the subgroup consisting of all elements of $G$ that have finite order. Prove that every nonidentity element in $G / H$ has infinite order.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:40

Problem 53

Determine all subgroups of $\mathbf{R}^{*}$ that have finite index.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:31

Problem 54

Let $G=\{\pm 1, \pm i, \pm j, \pm k\}$, where $i^{2}=j^{2}=k^{2}=-1,-i=(-1) i$,
$1^{2}=(-1)^{2}=1, i j=-j i=k, j k=-k j=i$, and $k i=-i k=j$
a. Show that $H=\{1,-1\} \triangleleft G$.
b. Construct the Cayley table for $G / H$. Is $G / H$ isomorphic to $Z_{4}$ or $Z_{2} \oplus Z_{2} ?$
(The rules involving $i, j$, and $k$ can be remembered by using the circle below.
Going clockwise, the product of two consecutive elements is the third one. The same is true for going counterclockwise, except that we obtain the negative of the third element. This group is called the quaternions. It was invented by William Hamilton in $1843 .$ The quaternions are used to describe rotations in three-dimensional space, and they are used in physics. The quaternions can be used to extend the complex numbers in a natural way).

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
03:08

Problem 55

In $D_{4}$, let $K=\left\{R_{0}, D\right\}$ and let $L=\left\{R_{0}, D, D^{\prime}, R_{180}\right\} .$ Show that $K \triangleleft$ $L \triangleleft D_{4}$, but that $K$ is not normal in $D_{4}$. (Normality is not transitive.)

Mj Santos
Mj Santos
Numerade Educator
02:05

Problem 56

Show that the intersection of two normal subgroups of $G$ is a normal subgroup of $G$. Generalize.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:07

Problem 57

Give an example of subgroups $H$ and $K$ of a group $G$ such that $H K$ is not a subgroup of $G$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
00:59

Problem 58

If $N$ and $M$ are normal subgroups of $G$, prove that $N M$ is also a normal subgroup of $G$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
00:59

Problem 59

Let $N$ be a normal subgroup of a group $G .$ If $N$ is cyclic, prove that every subgroup of $N$ is also normal in $G$. (This exercise is referred to in Chapter 24.)

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
00:59

Problem 60

Without looking at inner automorphisms of $D_{n}$, determine the number of such automorphisms.

Ronald Prasad
Ronald Prasad
Numerade Educator
00:59

Problem 61

Let $H$ be a normal subgroup of a finite group $G$ and let $x \in G .$ If $\operatorname{gcd}(|x|,|G / H|)=1$, show that $x \in H$. (This exercise is referred to in Chapter 25.)

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:07

Problem 62

Let $G$ be a group and let $G^{\prime}$ be the subgroup of $G$ generated by the $\operatorname{set} S=\left\{x^{-1} y^{-1} x y \mid x, y \in G\right\}$
a. Prove that $G^{\prime}$ is normal in $G$.
b. Prove that $G / G^{\prime}$ is Abelian.
c. If $G / N$ is Abelian, prove that $G^{\prime} \leq N$.
d. Prove that if $H$ is a subgroup of $G$ and $G^{\prime} \leq H$, then $H$ is normal in $G$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:57

Problem 63

Prove that the group $\mathbf{C}^{*} / \mathbf{R}^{*}$ has infinite order.

Wendi Zhao
Wendi Zhao
Numerade Educator
01:40

Problem 64

Suppose that a group $G$ has a subgroup of order $n$. Prove that the intersection of all subgroups of $G$ of order $n$ is a normal subgroup of $G$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
04:45

Problem 65

If $G$ is non-Abelian, show that $\operatorname{Aut}(G)$ is not cyclic.

Mengchun Cai
Mengchun Cai
Numerade Educator
01:40

Problem 66

Let $|G|=p^{n} m$, where $p$ is prime and $\operatorname{gcd}(p, m)=1$. Suppose that $H$ is a normal subgroup of $G$ of order $p^{n} .$ If $K$ is a subgroup of $G$ of order $p^{k}$, show that $K \subseteq H$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
00:59

Problem 67

Suppose that $H$ is a normal subgroup of a finite group $G$. If $G / H$ has an element of order $n$, show that $G$ has an element of order $n$. Show, by example, that the assumption that $G$ is finite is necessary.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:57

Problem 68

Prove that $A_{4}$ is the only subgroup of $S_{4}$ of order 12 .

Wendi Zhao
Wendi Zhao
Numerade Educator
01:53

Problem 69

If $|G|=30$ and $|Z(G)|=5$, what is the structure of $G / Z(G)$ ? What is the structure of $G / Z(G)$ if $|Z(G)|=3$ ? Generalize to the case that $|G|=2 p q$ where $p$ and $q$ are distinct odd primes.

Gaurav Kalra
Gaurav Kalra
Numerade Educator
01:07

Problem 70

If $H$ is a normal subgroup of $G$ and $|H|=2$, prove that $H$ is contained in the center of $G$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
06:02

Problem 71

Prove that $A_{5}$ cannot have a normal subgroup of order 2 .

Ely Crowder
Ely Crowder
Numerade Educator
01:02

Problem 72

Let $G$ be a group and $H$ an odd-order subgroup of $G$ of index 2 . Show that $H$ contains every element of $G$ of odd order.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator