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Contemporary Abstract Algebra

Joseph Gallian

Chapter 3

Finite Groups; Subgroups - all with Video Answers

Educators

Chapter Questions

00:27

Problem 1

For each group in the following list, find the order of the group and the order of each element in the group. What relation do you see between the orders of the elements of a group and the order of the group?
$$
Z_{12}, \quad U(10), \quad U(12), \quad U(20), \quad D_{4}
$$

Sam Limsuwannarot
Sam Limsuwannarot
Numerade Educator
30:00

Problem 2

Let $Q$ be the group of rational numbers under addition and let $Q^{*}$ be the group of nonzero rational numbers under multiplication. In $Q$, list the elements in $\left\langle\frac{1}{2}\right\rangle .$ In $Q^{*}$, list the elements in $\left\langle\frac{1}{2}\right\rangle$.

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
01:15

Problem 3

Let $Q$ and $Q^{*}$ be as in Exercise 2 . Find the order of each element in $Q$ and in $Q^{*}$.

JH
J Hardin
Numerade Educator
01:13

Problem 4

Prove that in any group, an element and its inverse have the same order.

Wendi Zhao
Wendi Zhao
Numerade Educator
00:58

Problem 5

Without actually computing the orders, explain why the two elements in each of the following pairs of elements from $Z_{30}$ must have the same order: $\{2,28\},\{8,22\}$. Do the same for the following pairs of elements from $U(15):\{2,8\},\{7,13\}$.

Lottie Adams
Lottie Adams
Numerade Educator
01:31

Problem 6

In the group $Z_{12}$, find $|a|,|b|$, and $|a+b|$ for each case.
a. $a=6, b=2$
b. $a=3, b=8$
c. $a=5, b=4$

Anthony Ramos
Anthony Ramos
Numerade Educator
01:16

Problem 7

If $a, b$, and $c$ are group elements and $|a|=6,|b|=7$, express $\left(a^{4} c^{-2} b^{4}\right)^{-1}$ without using negative exponents.

Alayna Handiak
Alayna Handiak
Numerade Educator
02:02

Problem 8

What can you say about a subgroup of $D_{3}$ that contains $R_{240}$ and a reflection $F ?$ What can you say about a subgroup of $D_{3}$ that contains two reflections?

Gopesh Vishwakarma
Gopesh Vishwakarma
Numerade Educator
02:05

Problem 9

What can you say about a subgroup of $D_{4}$ that contains $R_{270}$ and a reflection? What can you say about a subgroup of $D_{4}$ that contains $H$ and $D ?$ What can you say about a subgroup of $D_{4}$ that contains $H$ and $V$ ?

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:30

Problem 10

How many subgroups of order 4 does $D_{4}$ have?

Nick Johnson
Nick Johnson
Numerade Educator
00:46

Problem 11

Determine all elements of finite order in $R^{*}$, the group of nonzero real numbers under multiplication.

Charles Machakwa
Charles Machakwa
Numerade Educator
01:53

Problem 12

If $a$ and $b$ are group elements and $a b \neq b a$, prove that $a b a \neq e$.

Wendi Zhao
Wendi Zhao
Numerade Educator
01:02

Problem 13

Suppose that $H$ is a nonempty subset of a group $G$ that is closed under the group operation and has the property that if $a$ is not in $H$ then $a^{-1}$ is not in $H$. Is $H$ a subgroup?

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:03

Problem 14

Let $G$ be the group of polynomials under addition with coefficients from $Z_{10} .$ Find the orders of $f(x)=7 x^{2}+5 x+4, g(x)=4 x^{2}+8 x$ $+6$, and $f(x)+g(x)=x^{2}+3 x .$ If $h(x)=a_{n} x^{n}+a_{n}-1 x^{n-1}+\cdots$
$+a_{0}$ belongs to $G$, determine $|h(x)|$ given that $\operatorname{gcd}\left(a_{1}, a_{2}, \ldots, a_{n}\right)=1$; $\operatorname{gcd}\left(a_{1}, a_{2}, \ldots, a_{n}\right)=2 ; \operatorname{gcd}\left(a_{1}, a_{2}, \ldots, a_{n}\right)=5 ;$ and $\operatorname{gcd}\left(a_{1}, a_{2}, \ldots\right.$
$\left.a_{n}\right)=10$

Hast Aggarwal
Hast Aggarwal
Numerade Educator
01:57

Problem 15

If $a$ is an element of a group $G$ and $|a|=7$, show that $a$ is the cube of some element of $G$.

Wendi Zhao
Wendi Zhao
Numerade Educator
01:02

Problem 16

Suppose that $H$ is a nonempty subset of a group $G$ with the property that if $a$ and $b$ belong to $H$ then $a^{-1} b^{-1}$ belongs to $H .$ Prove or disprove that this is enough to guarantee that $H$ is a subgroup of $G$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
06:02

Problem 17

Prove that if an Abelian group has more than three elements of order 2, then it has at least 7 elements of order $2 .$ Find an example that shows this is not true for non-Abelian groups.

Ely Crowder
Ely Crowder
Numerade Educator
01:26

Problem 18

Suppose that $a$ is a group element and $a^{6}=e$. What are the possibilities for $|a|$ ? Provide reasons for your answer.

Aadit Sharma
Aadit Sharma
Numerade Educator
01:57

Problem 19

If $a$ is a group element and $a$ has infinite order, prove that $a^{m} \neq a^{n}$ when $m \neq n$.

Wendi Zhao
Wendi Zhao
Numerade Educator
01:57

Problem 20

Let $x$ belong to a group. If $x^{2} \neq e$ and $x^{6}=e$, prove that $x^{4} \neq e$ and $x^{5} \neq e$. What can we say about the order of $x$ ?

Wendi Zhao
Wendi Zhao
Numerade Educator
00:55

Problem 21

Show that if $a$ is an element of a group $G$, then $|a| \leq|G|$.

AG
Ankit Gupta
Numerade Educator
01:38

Problem 22

Show that $U(14)=\langle 3\rangle=\langle 5\rangle .$ [Hence, $U(14)$ is cyclic.] Is $U(14)=\langle 11\rangle ?$

Abhijith V
Abhijith V
Numerade Educator
01:18

Problem 23

Show that $U(20) \neq\langle k\rangle$ for any $k$ in $U(20) .$ [Hence, $U(20)$ is not cyclic.]

Tyler Moulton
Tyler Moulton
Numerade Educator
00:59

Problem 24

Suppose $n$ is an even positive integer and $H$ is a subgroup of $Z_{n}$. Prove that either every member of $H$ is even or exactly half of the members of $H$ are even.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:02

Problem 25

Prove that for every subgroup of $D_{n}$, either every member of the subgroup is a rotation or exactly half of the members are rotations.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
08:50

Problem 26

Prove that a group with two elements of order 2 that commute must have a subgroup of order $4 .$

Ely Crowder
Ely Crowder
Numerade Educator
03:09

Problem 27

For every even integer $n$, show that $D_{n}$ has a subgroup of order 4 .

Julian Wong
Julian Wong
Numerade Educator
01:42

Problem 28

Suppose that $H$ is a proper subgroup of $Z$ under addition and $H$ contains 18,30 , and 40. Determine $H$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:28

Problem 29

Suppose that $H$ is a proper subgroup of $Z$ under addition and that $H$ contains 12,30, and 54 . What are the possibilities for $H$ ?

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:57

Problem 30

Prove that the dihedral group of order 6 does not have a subgroup of order 4 .

Wendi Zhao
Wendi Zhao
Numerade Educator
01:40

Problem 31

For each divisor $k>1$ of $n$, let $U_{k}(n)=\{x \in U(n) \mid x \bmod k=1\}$. $\left[\right.$ For example, $U_{3}(21)=\{1,4,10,13,16,19\}$ and $\left.U_{7}(21)=\{1,8\} .\right]$
List the elements of $U_{4}(20), U_{5}(20), U_{5}(30)$, and $U_{10}(30) .$ Prove that $U_{k}(n)$ is a subgroup of $U(n)$. Let $H=\{x \in U(10) \mid x \bmod 3=1\} .$ Is $\underline{H}$ a subgroup of $U(10) ?$ (This exercise is referred to in Chapter 8.)

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:05

Problem 32

If $H$ and $K$ are subgroups of $G$, show that $H \cap K$ is a subgroup of $G$. (Can you see that the same proof shows that the intersection of any number of subgroups of $G$, finite or infinite, is again a subgroup of $G ?)$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:05

Problem 33

Let $G$ be a group. Show that $Z(G)=\cap_{a \in G} C(a) .$ [This means the intersection of all subgroups of the form $C(a) .]$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:58

Problem 34

Let $G$ be a group, and let $a \in G$. Prove that $C(a)=C\left(a^{-1}\right)$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:05

Problem 35

For any group element $a$ and any integer $k$, show that $C(a) \subseteq C\left(a^{k}\right)$. Use this fact to complete the following statement: "In a group, if $x$ commutes with $a$, then $\ldots .$ " Is the converse true?

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
11:16

Problem 36

Complete the partial Cayley group table given below.

Ely Crowder
Ely Crowder
Numerade Educator
01:16

Problem 37

Suppose $G$ is the group defined by the following Cayley table.

Wendi Zhao
Wendi Zhao
Numerade Educator
01:14

Problem 38

If $a$ and $b$ are distinct group elements, prove that either $a^{2} \neq b^{2}$ or $a^{3} \neq b^{3}$

Edward Downes
Edward Downes
Numerade Educator
02:05

Problem 39

Let $S$ be a subset of a group and let $H$ be the intersection of all subgroups of $G$ that contain $S .$
a. Prove that $\langle S\rangle=H$.
b. If $S$ is nonempty, prove that $\langle S\rangle=\left\{s_{1}^{n} 1 s_{2}^{n} 2 \ldots s_{m}^{n_{m}} \mid m \geq 1, s_{i} \in S\right.$,
$\left.n_{i} \in Z\right\} .$ (The $s_{i}$ terms need not be distinct.)

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:40

Problem 40

In the group $Z$, find
a. $\langle 8,14\rangle$;
b. $\langle 8,13\rangle$;
c. $\langle 6,15\rangle$;
d. $\langle m, n\rangle$;
e. $\langle 12,18,45\rangle$.
In each part, find an integer $k$ such that the subgroup is $\langle k\rangle$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:18

Problem 41

Prove Theorem $3.6$.

Nick Johnson
Nick Johnson
Numerade Educator
01:02

Problem 42

If $H$ is a subgroup of $G$, then by the centralizer $C(H)$ of $H$ we mean the set $\{x \in G \mid x h=h x$ for all $h \in H\} .$ Prove that $C(H)$ is a subgroup of $G$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
View

Problem 43

Must the centralizer of an element of a group be Abelian?

Nick Johnson
Nick Johnson
Numerade Educator
View

Problem 44

Must the center of a group be Abelian?

Nick Johnson
Nick Johnson
Numerade Educator
00:59

Problem 45

Let $G$ be an Abelian group with identity $e$ and let $n$ be some fixed integer. Prove that the set of all elements of $G$ that satisfy the equation $x^{n}=e$ is a subgroup of $G$. Give an example of a group $G$ in which the set of all elements of $G$ that satisfy the equation $x^{2}=e$ does not form a subgroup of $G$. (This exercise is referred to in Chapter 11.)

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:30

Problem 46

Suppose $a$ belongs to a group and $|a|=5$. Prove that $C(a)=C\left(a^{3}\right)$. Find an element $a$ from some group such that $|a|=6$ and $C(a) \neq$ $C\left(a^{3}\right)$

Angelo Rendina
Angelo Rendina
Numerade Educator
03:02

Problem 47

Let $G$ be the set of all polynomials with coefficients from the set $\{0,1,2,3\} .$ We can make $G$ a group under addition by adding the polynomials in the usual way, except that we use modulo 4 to combine the coefficients. With this group operation, determine the orders of the elements of $G .$ Determine a necessary and sufficient condition for an element of $G$ to have order 2 .

Allison Knapp
Allison Knapp
Numerade Educator
01:31

Problem 48

In each case, find elements $a$ and $b$ from a group such that $|a|=$ $|b|=2$.
a. $|a b|=3$
b. $|a b|=4$
c. $|a b|=5$
Can you see any relationship among $|a|,|b|$, and $|a b|$ ?

Anthony Ramos
Anthony Ramos
Numerade Educator
01:44

Problem 49

Suppose a group contains elements $a$ and $b$ such that $|a|=4$, $|b|=2$, and $a^{3} b=b a$. Find $|a b|$.

Anthony Ramos
Anthony Ramos
Numerade Educator
01:28

Problem 50

Suppose $a$ and $b$ are group elements such that $|a|=2, b \neq e$, and $a b a=b^{2} .$ Determine $|b| .$

Harshita Goel
Harshita Goel
Numerade Educator
04:33

Problem 51

Let $a$ be a group element of order $n$, and suppose that $d$ is a positive divisor of $n$. Prove that $\left|a^{d}\right|=n / d$.

James Chok
James Chok
Numerade Educator
06:40

Problem 52

Consider the elements $A=\left[\begin{array}{rr}0 & 1 \\ 1 & 0\end{array}\right]$ and $B=\left[\begin{array}{rr}0 & 1 \\ -1 & -1\end{array}\right]$ from
$S L(2, \mathbf{R}) .$ Find $|\mathrm{Al},| \mathrm{B} \mid$, and $|\mathrm{AB}|$. Does your answer surprise you?

David Mccaslin
David Mccaslin
Numerade Educator
02:57

Problem 53

Consider the element $A=\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right]$ in $S L(2, \mathbf{R}) .$ What is the order of $A ?$ If we view $A=\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right]$ as a member of $S L\left(2, Z_{p}\right)(p$ is a prime), what is the order of $A$ ?

Bryan Lynn
Bryan Lynn
Numerade Educator
02:19

Problem 54

For any positive integer $n$ and any angle $\theta$, show that in the group $S L(2, \mathbf{R})$
$$
\left[\begin{array}{l}
\cos \theta-\sin \theta \\
\sin \theta & \cos \theta
\end{array}\right]^{n}=\left[\begin{array}{c}
\cos n \theta-\sin n \theta \\
\sin n \theta & \cos n \theta
\end{array}\right] .
$$
Use this formula to find the order of
$$
\left[\begin{array}{lr}
\cos 60^{\circ}-\sin 60^{\circ} \\
\sin 60^{\circ} & \cos 60^{\circ}
\end{array}\right] \text { and }\left[\begin{array}{l}
\cos \sqrt{2}^{\circ}-\sin \sqrt{2}^{\circ} \\
\sin \sqrt{2^{\circ}} & \cos \sqrt{2}^{\circ}
\end{array}\right]
$$
(Geometrically, $\left[\begin{array}{rr}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{array}\right]$ represents a rotation of the plane
$\theta$ degrees.)

Tanishq Gupta
Tanishq Gupta
Numerade Educator
01:57

Problem 55

Let $G$ be the symmetry group of a circle. Show that $G$ has elements of every finite order as well as elements of infinite order.

Wendi Zhao
Wendi Zhao
Numerade Educator
01:14

Problem 56

Let $x$ belong to a group and $|x|=6$. Find $\left|x^{2}\right|,\left|x^{3}\right|,\left|x^{4}\right|$, and $\left|x^{5}\right|$. Let $y$ belong to a group and $|y|=9$. Find $\left|y^{i}\right|$ for $i=2,3, \ldots, 8$. Do these examples suggest any relationship between the order of the power of an element and the order of the element?

Teresa Fuston
Teresa Fuston
Numerade Educator
05:10

Problem 57

$D_{4}$ has seven cyclic subgroups. List them.

Ely Crowder
Ely Crowder
Numerade Educator
03:46

Problem 58

$U(15)$ has six cyclic subgroups. List them.

Mengchun Cai
Mengchun Cai
Numerade Educator
View

Problem 59

Prove that a group of even order must have an element of order $2 .$

Nick Johnson
Nick Johnson
Numerade Educator
01:40

Problem 60

Suppose $G$ is a group that has exactly eight elements of order 3 . How many subgroups of order 3 does $G$ have?

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
00:59

Problem 61

Let $H$ be a subgroup of a finite group $G$. Suppose that $g$ belongs to $G$ and $n$ is the smallest positive integer such that $g^{n} \in H$. Prove that $n$ divides $|g|$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
15:38

Problem 62

Compute the orders of the following groups.
a. $U(3), U(4), U(12)$
b. $U(5), U(7), U(35)$
c. $U(4), U(5), U(20)$
d. $U(3), U(5), U(15)$
On the basis of your answers, make a conjecture about the relationship among $|U(r)|,|U(s)|$, and $|U(r s)|$.

Abigail Martyr
Abigail Martyr
Numerade Educator
00:59

Problem 63

Let $\mathbf{R}^{*}$ be the group of nonzero real numbers under multiplication and let $H=\left\{x \in \mathbf{R}^{*} \mid x^{2}\right.$ is rational $\} .$ Prove that $H$ is a subgroup of $\mathbf{R}^{*}$. Can the exponent 2 be replaced by any positive integer and still have $H$ be a subgroup?

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:34

Problem 64

Compute $|U(4)|,|U(10)|$, and $|U(40)| .$ Do these groups provide a counterexample to your answer to Exercise $62 ?$ If so, revise your conjecture.

Carlos Pinilla
Carlos Pinilla
Numerade Educator
00:14

Problem 65

Find a cyclic subgroup of order 4 in $U(40)$.

IC
Isaac Chiu
Numerade Educator
00:14

Problem 66

Find a noncyclic subgroup of order 4 in $U(40)$.

IC
Isaac Chiu
Numerade Educator
02:55

Problem 67

Let $G=\left\{\left[\begin{array}{ll}a & b \\ c & d\end{array}\right] \mid a, b, c, d \in Z\right\}$ under addition. Let $H=$
$\left\{\left[\begin{array}{ll}a & b \\ c & d\end{array}\right] \in G \mid a+b+c+d=0\right\}$. Prove that $H$ is a subgroup of $G$. What if 0 is replaced by $1 ?$

Nick Johnson
Nick Johnson
Numerade Educator
01:07

Problem 68

Let $H=\{A \in G L(2, \mathbf{R}) \mid \operatorname{det} A$ is an integer power of 2$\} .$ Show that $H$ is a subgroup of $G L(2, \mathbf{R})$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:07

Problem 69

Let $H$ be a subgroup of $\mathbf{R}$ under addition. Let $K=\left\{2^{a} \mid a \in H\right\}$. Prove that $K$ is a subgroup of $\mathbf{R}^{*}$ under multiplication.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:02

Problem 70

Let $G$ be a group of functions from $\mathbf{R}$ to $\mathbf{R}^{*}$, where the operation of $G$ is multiplication of functions. Let $H=\{f \in G \mid f(2)=1\}$. Prove that $H$ is a subgroup of $G .$ Can 2 be replaced by any real number?

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:02

Problem 71

Let $G=G L(2, \mathbf{R})$ and $H=\left\{\left[\begin{array}{ll}a & 0 \\ 0 & b\end{array}\right] \mid a\right.$ and $b$ are nonzero integers $\}$ under the operation of matrix multiplication. Prove or disprove that $H$ is a subgroup of $G L(2, \mathbf{R})$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:02

Problem 72

Let $H=\{a+b i \mid a, b \in \mathbf{R}, a b \geq 0\} .$ Prove or disprove that $H$ is a subgroup of $\mathbf{C}$ under addition.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:02

Problem 73

Let $H=\left\{a+b i \mid a, b \in \mathbf{R}, a^{2}+b^{2}=1\right\} .$ Prove or disprove that $H$ is a subgroup of $\mathbf{C}^{*}$ under multiplication. Describe the elements of $H$ geometrically.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:40

Problem 74

Let $G$ be a finite Abelian group and let $a$ and $b$ belong to $G$. Prove that the set $\langle a, b\rangle=\left\{a^{i} b^{j} \mid i, j \in \mathrm{Z}\right\}$ is a subgroup of $G .$ What can you say about $|\langle a, b\rangle|$ in terms of $|a|$ and $|b|$ ?

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:40

Problem 75

Let $H$ be a subgroup of a group $G$. Prove that the set $H Z(G)=$ $\{h z \mid h \in H, z \in Z(G)\}$ is a subgroup of $G .$ This exercise is referred to in this chapter.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:07

Problem 76

Let $G$ be a group and $H$ a subgroup. For any element $g$ of $G$, define $g H=\{g h \mid h \in H\}$. If $G$ is Abelian and $g$ has order 2, show that the set $K=H \cup g H$ is a subgroup of $G$. Is your proof valid if we drop the assumption that $G$ is Abelian and let $K=Z(G) \cup g Z(G)$ ?

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
03:56

Problem 77

Let $a$ belong to a group and $|a|=m$. If $n$ is relatively prime to $m$, show that $a$ can be written as the $n$ th power of some element in the group.

Sriparna Bhattacharjee
Sriparna Bhattacharjee
Numerade Educator
14:44

Problem 78

Let $F$ be a reflection in the dihedral group $D_{n}$ and $R$ a rotation in $D_{n} .$ Determine $C(F)$ when $n$ is odd. Determine $C(F)$ when $n$ is even. Determine $C(R)$.

Chris Trentman
Chris Trentman
Numerade Educator
02:01

Problem 79

Let $G=G L(2, \mathbf{R})$.
a. Find $C\left(\left[\begin{array}{ll}1 & 1 \\ 1 & 0\end{array}\right]\right)$.
b. Find $C\left(\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]\right)$.
c. Find $Z(G)$.

Gregory Higby
Gregory Higby
Numerade Educator
01:57

Problem 80

Let $G$ be a finite group with more than one element. Show that $G$ has an element of prime order.

Wendi Zhao
Wendi Zhao
Numerade Educator