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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 $\left(\frac{1}{2}\right)$. In $Q^{*}$, list the elements in $\left\langle\frac{1}{2}\right.$.

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. $a=5, b=4$
Do you see any relationship between $|a|,|b|$, and $|a+b|$ ?

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
01:09

Problem 12

Complete the statement "A group element $x$ is its own inverse if and only if $|x|=$ $-. "$

James Kiss
James Kiss
Numerade Educator
00:43

Problem 13

For any group elements $a$ and $x$, prove that $\left|x a x^{-1}\right|=|a|$. This exercise is referred to in Chapter $24 .$

Mike Gaerlan
Mike Gaerlan
Numerade Educator
View

Problem 14

Prove that if $a$ is the only element of order 2 in a group, then $a$ lies in the center of the group.

Nick Johnson
Nick Johnson
Numerade Educator
01:18

Problem 15

(1969 Putnam Competition) Prove that no group is the union of two proper subgroups. Does the statement remain true if "two" is replaced by "three"?

Wendi Zhao
Wendi Zhao
Numerade Educator
02:30

Problem 16

Let $G$ be the group of symmetries of a circle and $R$ be a rotation of the circle of $\sqrt{2}$ degrees. What is $|R|$ ?

Wendi Zhao
Wendi Zhao
Numerade Educator
01:40

Problem 17

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 $H$ a subgroup of $U(10) ?$ (This exercise is referred to in Chapter 8.)

Varsha Aggarwal
Varsha Aggarwal
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
00:45

Problem 20

For any group elements $a$ and $b$, prove that $\mathrm{lab} \mid=\mathrm{lbal}$.

Nick Johnson
Nick Johnson
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
00:59

Problem 25

Let $n$ be a positive even integer and let $H$ be a subgroup of $Z_{n}$ of odd order. Prove that every member of $H$ is an even integer.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:02

Problem 26

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
01:02

Problem 27

Let $H$ be a subgroup of $D_{n}$ of odd order. Prove that every member of $H$ is a rotation.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
08:50

Problem 28

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 29

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

Julian Wong
Julian Wong
Numerade Educator
01:42

Problem 30

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 31

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
01:42

Problem 32

Suppose that $H$ is a subgroup of $Z$ under addition and that $H$ contains $2^{50}$ and $3^{50}$. What are the possibilities for $H$ ?

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:57

Problem 33

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

Wendi Zhao
Wendi Zhao
Numerade Educator
02:05

Problem 34

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 35

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 36

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 37

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
01:02

Problem 38

Let $G$ be an Abelian group and $H=\{x \in G|| x \mid$ is odd $\} .$ Prove that $H$ is a subgroup of $G .$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:07

Problem 39

Let $G$ be an Abelian group and $H=\{x \in G \mid \mathrm{xl}$ is 1 or even $\}$. Give an example to show that $H$ need not be a subgroup of $G$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:14

Problem 40

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 41

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 $\langle S\rangle=\left\{s_{1}^{n_{1}} s_{2}^{H_{2}} \cdots s_{m}^{n_{m}} \mid m \geq 1, s_{i} \in S,\right.$,
$\left.n_{i} \in Z\right\} .$ (The $s_{i}$ terms need not be distinct.)

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:40

Problem 42

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 43

Prove Theorem $3.6$.

Nick Johnson
Nick Johnson
Numerade Educator
01:02

Problem 44

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
01:35

Problem 45

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

Nick Johnson
Nick Johnson
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
00:59

Problem 47

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
01:31

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
01:51

Problem 49

Prove that a group of even order must have an odd number of elements of order $2 .$

James Chok
James Chok
Numerade Educator
06:40

Problem 50

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 $\lfloor A|,| B \mid$,, and $\ A B \mid$. Does your answer surprise you?

David Mccaslin
David Mccaslin
Numerade Educator
02:58

Problem 51

Let $a$ be a group element of order $n$, and suppose that $d$ is a positive divisor of $n$. Prove that $1 a^{d} \underline{l}=n / d$.

Carson Merrill
Carson Merrill
Numerade Educator
01:14

Problem 52

Give an example of elements $a$ and $b$ from a group such that $a$ has finite order, $b$ has infinite order and $a b$ has finite order.

Doruk Isik
Doruk Isik
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 $\operatorname{SL}(2, \mathbf{R})$
$$
\left[\begin{array}{rr}
\cos \theta-\sin \theta \\
\sin \theta & \cos \theta
\end{array}\right]^{n}=\left[\begin{array}{cc}
\cos n \theta-\sin n \theta \\
\sin n \theta & \cos n \theta
\end{array}\right] .
$$
Use this formula to find the order of
$$
\left[\begin{array}{rr}
\cos 60^{\circ}-\sin 60^{\circ} \\
\sin 60^{\circ} & \cos 60^{\circ}
\end{array}\right] \text { and }\left[\begin{array}{l}
\cos \sqrt{2}^{\circ}-\sin \sqrt{2}^{\circ} \\
\sin \sqrt{2}^{\circ} & \cos \sqrt{2}^{\circ}
\end{array}\right]
$$
(Geometrically, $\left[\begin{array}{lr}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{array}\right]$ 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
02:42

Problem 56

In the group $\mathbf{R}^{*}$ find elements $a$ and $b$ such that $|a|=\infty,|b|=\infty$ and $|a b|=2$.

Brandon Collins
Brandon Collins
Numerade Educator
01:16

Problem 57

Let $G$ be the symmetry group of a circle. Explain why $G$ contains $D_{n}$ for all $n$.

Wendi Zhao
Wendi Zhao
Numerade Educator
06:02

Problem 58

Prove that the subset of elements of finite order in an Abelian group forms a subgroup. (This subgroup is called the torsion subgroup.) Is the same thing true for non-Abelian groups?

Ely Crowder
Ely Crowder
Numerade Educator
00:59

Problem 59

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 60

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 61

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
01:13

Problem 62

Compute $|U(4)|,|U(10)|$, and $|U(40)| .$ Do these groups provide a counterexample to your answer to Exercise 60 ? If so, revise your conjecture.

James Kiss
James Kiss
Numerade Educator
00:14

Problem 63

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

IC
Isaac Chiu
Numerade Educator
View

Problem 64

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

Nick Johnson
Nick Johnson
Numerade Educator
02:55

Problem 65

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 66

Let $H=\{A \in G L(2, \mathbf{R}) \mid$ 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 67

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 68

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 69

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 70

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 71

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 72

Let $G$ be a finite Abelian group and let $a$ and $b$ belong to $G$. Prove that the set $\langle a, b\rangle=\left\{a^{i} b^{j} \mid i, j \in Z\right\}$ is a subgroup of $G .$ What can you say about $1 / a, b]$ in terms of $|a|$ and $|b| ?$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:40

Problem 73

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
01:52

Problem 74

If $H$ and $K$ are nontrivial subgroups of the rational numbers under addition, prove that $H \cap K$ is nontrivial.

Mengchun Cai
Mengchun Cai
Numerade Educator
01:02

Problem 75

Let $H$ be a nontrival subgroup of the group of rational numbers under addition. Prove that $H$ has a nontrivial proper subgroup.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
00:46

Problem 76

Prove that a group of order $n$ greater than 2 cannot have a subgroup of order $n-1$.

Nick Johnson
Nick Johnson
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
01:57

Problem 78

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