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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 111 , 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
01:07

Problem 11

Let $G=Z_{4} \oplus U(4), H=\langle(2,3)\rangle$, and $K=\langle(2,1)\rangle .$ Show that $G / H$
is not isomorphic to $G / K .$ (This shows that $H \approx K$ does not imply that $G / H \approx G / K .$ )

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:35

Problem 12

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

Nick Johnson
Nick Johnson
Numerade Educator
View

Problem 13

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

Nick Johnson
Nick Johnson
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
01:11

Problem 19

What is the order of the factor group $\left(Z_{10} \oplus U(10)\right) /\langle(2,9)\rangle$ ?

Nez Nikoo
Nez Nikoo
Numerade Educator
00:15

Problem 20

Construct the Cayley table for $U(20) / U_{5}(20)$.

Sahil Patel
Sahil Patel
Numerade Educator
View

Problem 21

Prove that an Abelian group of order 33 is cyclic.

Nick Johnson
Nick Johnson
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
01:02

Problem 23

Determine the order of $(Z \oplus Z) /\langle(4,2)\rangle .$ Is the group cyclic?

Hunza Gilgit
Hunza Gilgit
Numerade Educator
02:28

Problem 24

The group $\left(Z_{4} \oplus Z_{12}\right) /\langle(2,2)\rangle$ 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 $H=\{1,31\}$. 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
05:10

Problem 26

Let $G$ be the group of quaternions given by the table in Exercise 4 of the Supplementary Exercises for Chapters $1-4$, and let $H$ be the subgroup $\left\{e, a^{2}\right\}$. Is $G / H$ isomorphic to $Z_{4}$ or $Z_{2} \oplus Z_{2} ?$

Ely Crowder
Ely Crowder
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
02:45

Problem 29

Prove that $A_{4} \oplus Z_{3}$ has no subgroup of order 18 .

Uma Kumari
Uma Kumari
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
08:50

Problem 32

Prove that $D_{4}$ cannot be expressed as an internal direct product of two proper subgroups.

Ely Crowder
Ely Crowder
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
01:40

Problem 34

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
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
01:02

Problem 38

Let $H$ be a normal subgroup of $G$ and let $a$ belong to $G$. If the element $a H$ has order 3 in the group $G / H$ and $|H|=10$, what are the possibilities for the order of $a$ ?

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:02

Problem 39

. If $H$ is a normal subgroup of a group $G$, prove that $C(H)$, the centralizer of $H$ in $G$, is a normal subgroup of $G$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:02

Problem 40

Let $\phi$ be an isomorphism from a group $G$ onto a group $\bar{G}$. Prove that if $H$ is a normal subgroup of $G$, then $\phi(H)$ is a normal subgroup of $\bar{G}$.

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 "nth 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
03:16

Problem 44

Observe from the table for $A_{4}$ given in Table $5.1$ on page 111 that the subgroup given in Example 9 of this chapter is the only subgroup of $A_{4}$ of order $4 .$ Why does this imply that this subgroup must be normal in $A_{4}$ ? Generalize this to arbitrary finite groups.

Rashmi Sinha
Rashmi Sinha
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
00:59

Problem 47

Suppose that $N$ is a normal subgroup of a finite group $G$ and $H$ is a subgroup of $G$. If $|G / N|$ is prime, prove that $H$ is contained in $N$ or that $N H=G$.

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
00:59

Problem 51

Let $N$ be a normal subgroup of $G$ and let $H$ be a subgroup of $G$. If $N$ is a subgroup of $H$, prove that $H / N$ is a normal subgroup of $G / N$ if and only if $H$ is a normal subgroup of $G$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:57

Problem 52

Let $G$ be an Abelian group and let $H$ be the subgroup consisting of all elements of $G$ that have finite order. (See Exercise 20 in the Supplementary Exercises for Chapters $1-4 .$ ) Prove that every nonidentity element in $G / H$ has infinite order.

Wendi Zhao
Wendi Zhao
Numerade Educator
11:16

Problem 53

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

Ely Crowder
Ely Crowder
Numerade Educator
05:10

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. Construct the Cayley table for $G$.
b. Show that $H=\{1,-1\}<G$.
c. 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.

Ely Crowder
Ely Crowder
Numerade Educator
02:18

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. Compare Exercise 4, Supplementary Exercises for Chapters 5-8.)

Vishal Parmar
Vishal Parmar
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
00:59

Problem 62

Let $G$ be a group and let $G^{\prime}$ be the subgroup of $G$ generated by the set $S=\left\{x^{-1} y^{-1} x y \mid x, y \in G\right\} .$ (See Exercise 3, Supplementary Exercises for Chapters $5-8$, for a more complete description of $G^{\prime} .$ )
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
00:59

Problem 63

. 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
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
00:59

Problem 68

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
01:52

Problem 69

In $D_{4}$, let $\mathscr{K}=\left\{R_{0}, H\right\} .$ Form an operation table for the cosets $\mathscr{K}$, $D \mathscr{H}, V \boldsymbol{K}$, and $D^{\prime} \mathscr{K}$. Is the result a group table? Does your answer contradict Theorem $9.2$ ?

WM
William Mead
Numerade Educator
01:57

Problem 70

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

Wendi Zhao
Wendi Zhao
Numerade Educator
04:06

Problem 71

If $|G|=30$ and $|Z(G)|=5$, what is the structure of $G / Z(G) ?$

Anupa Sharad Medhekar
Anupa Sharad Medhekar
Numerade Educator
01:07

Problem 72

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 73

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

Ely Crowder
Ely Crowder
Numerade Educator
01:02

Problem 74

Let $G$ be a finite group and let $H$ be an odd-order subgroup of $G$ of index 2 . Show that the product of all the elements of $G$ (taken in any order) cannot belong to $H$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:02

Problem 75

Let $G$ be a group and $p$ a prime. Suppose that $H=\left\{g^{p} \mid g \in G\right\}$ is a subgroup of $G$. Show that $H$ is normal and that every nonidentity element of $G / H$ has order $p$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:02

Problem 76

Suppose that $H$ is a normal subgroup of $G$. If $|H|=4$ and $g H$ has order 3 in $G / H$, find a subgroup of order 12 in $G$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:02

Problem 77

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
01:07

Problem 78

A proper subgroup $H$ of a group $G$ is called maximal if there is no subgroup $K$ such that $H \subset K \subset G$ (that is, there is no subgroup $K$ properly contained between $H$ and $G$ ). Show that $Z(G)$ is never a maximal subgroup of a group $G$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:02

Problem 79

Let $G$ be a group of order 100 that has exactly one subgroup of order $5 .$ Prove that it has a subgroup of order $10 .$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator