Contemporary Abstract Algebra

Joseph Gallian

Chapter 2 Groups - all with Video Answers

Educators

Chapter Questions

06:26

Problem 1

Which of the following binary operations are closed?
a. subtraction of positive integers
b. division of nonzero integers
c. function composition of polynomials with real coefficients
d. multiplication of $2 \times 2$ matrices with integer entries

Willis James

Numerade Educator

06:26

Problem 2

Which of the following binary operations are associative?
a. multiplication mod $n$
b. division of nonzero rationals
c. function composition of polynomials with real coefficients
d. multiplication of $2 \times 2$ matrices with integer entries

Willis James

Numerade Educator

06:26

Problem 3

Which of the following binary operations are commutative?
a. substraction of integers
b. division of nonzero real numbers
c. function composition of polynomials with real coefficients
d. multiplication of $2 \times 2$ matrices with real entries

Willis James

Numerade Educator

03:39

Problem 4

Which of the following sets are closed under the given operation?
a. $\{0,4,8,12\}$ addition $\bmod 16$
b. $\{0,4,8,12\}$ addition mod 15
c. $\{1,4,7,13\}$ multiplication mod 15
d. $\{1,4,5,7\}$ multiplication $\bmod 9$

Muhammad Nawaz

Numerade Educator

01:56

Problem 5

In each case, find the inverse of the element under the given operation.
a. 13 in $Z_{20}$
b. 13 in $U(14)$
c. $n-1$ in $U(n)(n>2)$
d. $3-2 i$ in $\mathbf{C}^{*}$, the group of nonzero complex numbers under multiplication

Kavin Shingala

Numerade Educator

02:15

Problem 6

In each case, perform the indicated operation.
a. $\operatorname{In}\left(\mathbf{C}^{*},(7+5 i)(-3+2 i)\right.$
b. $\operatorname{In} G L\left(2, Z_{13}\right), \operatorname{det}\left[\begin{array}{ll}7 & 4 \\ 1 & 5\end{array}\right]$
c. In $G L(2, R),\left[\begin{array}{ll}6 & 3 \\ 8 & 2\end{array}\right]^{-1}$
d. $\operatorname{In} G L\left(2, Z_{13}\right),\left[\begin{array}{ll}6 & 3 \\ 8 & 2\end{array}\right]^{-1}$

Megan Mcfarland

Numerade Educator

00:27

Problem 7

Give two reasons why the set of odd integers under addition is not a group.

Mitchell Cutler

Numerade Educator

01:35

Problem 10

Show that the group $G L(2, \mathbf{R})$ of Example 9 is non-Abelian by exhibiting a pair of matrices $A$ and $B$ in $G L(2, \mathbf{R})$ such that $A B \neq B A$.

Nick Johnson

Numerade Educator

00:40

Problem 11

Find the inverse of the element $\left[\begin{array}{ll}2 & 6 \\ 3 & 5\end{array}\right]$ in $G L\left(2, Z_{11}\right)$.

Charles Carter

Numerade Educator

00:47

Problem 12

Give an example of group elements $a$ and $b$ with the property that $a^{-1} b a \neq b$

Linh Vu

Numerade Educator

00:16

Problem 13

Translate each of the following multiplicative expressions into its additive counterpart. Assume that the operation is commutative.
a. $a^{2} b^{3}$
b. $a^{-2}\left(b^{-1} c\right)^{2}$
c. $\left(a b^{2}\right)^{-3} c^{2}=e$

Kristen Frankie

Numerade Educator

02:42

Problem 14

For group elements $a, b$, and $c$, express $(a b)^{3}$ and $\left(a b^{-2} c\right)^{-2}$ without parentheses.

Brandon Collins

Numerade Educator

00:59

Problem 15

Let $G$ be a group and let $H=\left\{x^{-1} \mid x \in G\right\}$. Show that $G=H$ as sets.

Varsha Aggarwal

Numerade Educator

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Problem 16

Show that the set $\{5,15,25,35\}$ is a group under multiplication modulo $40 .$ What is the identity element of this group? Can you see any relationship between this group and $U(8) ?$

Nick Johnson

Numerade Educator

01:13

Problem 17

(From the GRE Practice Exam) $^{*}$ Let $p$ and $q$ be distinct primes. Suppose that $H$ is a proper subset of the integers that is a group under addition that contains exactly three elements of the set $\{p, p+q,$, $\left.p q, p^{q}, q^{p}\right\} .$ Determine which of the following are the three elements in $H$.
a. $p q, p^{q}, q^{p}$
b. $p+q, p q, p^{q}$
c. $p, p+q, p q$
d. $p, p^{q}, q^{p}$
e. $p, p q, p^{q}$

Wen Zheng

Numerade Educator

00:37

Problem 18

List the members of $H=\left\{x^{2} \mid x \in D_{4}\right\}$ and $K=\left\{x \in D_{4} \mid x^{2}=e\right\}$.

Amy Jiang

Numerade Educator

03:22

Problem 19

Prove that the set of all $2 \times 2$ matrices with entries from $\mathbf{R}$ and determinant $+1$ is a group under matrix multiplication.

Adam Dehollander

Numerade Educator

02:21

Problem 20

For any integer $n>2$, show that there are at least two elements in $U(n)$ that satisfy $x^{2}=1$

Nick Johnson

Numerade Educator

01:32

Problem 21

An abstract algebra teacher intended to give a typist a list of nine integers that form a group under multiplication modulo $91 .$ Instead,

Narayan Hari

Numerade Educator

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Problem 22

Let $G$ be a group with the property that for any $x, y, z$ in the group, $x y=z x$ implies $y=z$. Prove that $G$ is Abelian. ("Left-right cancellation" implies commutativity.)

Nick Johnson

Numerade Educator

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Problem 23

(Law of Exponents for Abelian Groups) Let $a$ and $b$ be elements of an Abelian group and let $n$ be any integer. Show that $(a b)^{n}=a^{n} b^{n}$. Is this also true for non-Abelian groups?

Nick Johnson

Numerade Educator

03:38

Problem 24

(Socks-Shoes Property) Draw an analogy between the statement $(a b)^{-1}=b^{-1} a^{-1}$ and the act of putting on and taking off your socks and shoes. Find distinct nonidentity elements $a$ and $b$ from a non-Abelian group such that $(a b)^{-1}=a^{-1} b^{-1} .$ Find an example that shows that in a group, it is possible to have $(a b)^{-2} \neq b^{-2} a^{-2}$. What would be an appropriate name for the group property $(a b c)^{-1}=c^{-1} b^{-1} a^{-1} ?$

Wendi Zhao

Numerade Educator

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Problem 25

Prove that a group $G$ is Abelian if and only if $(a b)^{-1}=a^{-1} b^{-1}$ for all $a$ and $b$ in $G$.

Nick Johnson

Numerade Educator

02:53

Problem 26

Prove that in a group, $\left(a^{-1}\right)^{-1}=a$ for all $a$.

Wendi Zhao

Numerade Educator

01:59

Problem 27

For any elements $a$ and $b$ from a group and any integer $n$, prove that $\left(a^{-1} b a\right)^{n}=a^{-1} b^{n} a$.

Wendi Zhao

Numerade Educator

03:04

Problem 28

If $a_{1}, a_{2}, \ldots, a_{n}$ belong to a group, what is the inverse of $a_{1} a_{2} \cdots a_{n} ?$

Joshua Sieverding

Numerade Educator

00:54

Problem 29

The integers 5 and 15 are among a collection of 12 integers that form a group under multiplication modulo 56. List all 12 .

James Chok

Numerade Educator

01:52

Problem 30

Give an example of a group with 105 elements. Give two examples of groups with 44 elements.

Simon Kangoun

Numerade Educator

02:20

Problem 31

Prove that every group table is a Latin square $^{\dagger}$; that is, each element of the group appears exactly once in each row and each column.

William Mead

Numerade Educator

01:33

Problem 32

Construct a Cayley table for $U(12)$.

Victor Salazar

Numerade Educator

07:07

Problem 33

Suppose the table below is a group table. Fill in the blank entries.

Aparna Shakti

Numerade Educator

01:14

Problem 34

Prove that in a group, $(a b)^{2}=a^{2} b^{2}$ if and only if $a b=b a$.

Edward Downes

Numerade Educator

01:17

Problem 35

Let $a, b$, and $c$ be elements of a group. Solve the equation $a x b=c$ for $x$. Solve $a^{-1} x a=c$ for $x$.

Sriram Soundarrajan

Numerade Educator

00:58

Problem 36

Let $a$ and $b$ belong to a group $G$. Find an $x$ in $G$ such that $x a b x^{-1}=b a$.

Varsha Aggarwal

Numerade Educator

01:57

Problem 37

Let $G$ be a finite group. Show that the number of elements $x$ of $G$ such that $x^{3}=e$ is odd. Show that the number of elements $x$ of $G$ such that $x^{2} \neq e$ is even.

Wendi Zhao

Numerade Educator

01:16

Problem 38

Give an example of a group with elements $a, b, c, d$, and $x$ such that $a x b=c x d$ but $a b \neq c d$. (Hence "middle cancellation" is not valid in groups.)

Wendi Zhao

Numerade Educator

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Problem 39

Suppose that $G$ is a group with the property that for every choice of elements in $G, a x b=c x d$ implies $a b=c d$. Prove that $G$ is Abelian. ("Middle cancellation" implies commutativity.)

Nick Johnson

Numerade Educator

00:37

Problem 40

Find an element $X$ in $D_{4}$ such that $R_{90} V X H=D^{\prime}$.

Amy Jiang

Numerade Educator

02:19

Problem 41

Suppose $F_{1}$ and $F_{2}$ are distinct reflections in a dihedral group $D_{n}$. Prove that $F_{1} F_{2} \neq R_{0}$

John Nicolle

Numerade Educator

00:23

Problem 42

Suppose $F_{1}$ and $F_{2}$ are distinct reflections in a dihedral group $D_{n}$ such that $F_{1} F_{2}=F_{2} F_{1} .$ Prove that $F_{1} F_{2}=R_{180}$

Ashley High

Numerade Educator

05:38

Problem 43

Let $R$ be any fixed rotation and $F$ any fixed reflection in a dihedral group. Prove that $R^{k} F R^{k}=F$.

Mauricio Araiza Canizales

Numerade Educator

00:59

Problem 44

Let $R$ be any fixed rotation and $F$ any fixed reflection in a dihedral group. Prove that $F R^{k} F=R^{-k} .$ Why does this imply that $D_{n}$ is non-Abelian?

Varsha Aggarwal

Numerade Educator

30:00

Problem 46

Prove that the set of all rational numbers of the form $3^{m} 6^{n}$, where $m$ and $n$ are integers, is a group under multiplication.

Oswaldo Jiménez

Numerade Educator

04:45

Problem 47

Prove that if $G$ is a group with the property that the square of every element is the identity, then $G$ is Abelian. (This exercise is referred to in Chapter 26.)

Mengchun Cai

Numerade Educator

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Problem 48

Prove that the set of all $3 \times 3$ matrices with real entries of the form$\left[\begin{array}{lll}1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1\end{array}\right]$

Anecia Mcmurrin-Bala

Numerade Educator

09:06

Problem 49

Prove the assertion made in Example 20 that the set $\{1,2, \ldots,$, $n-1\}$ is a group under multiplication modulo $n$ if and only if $n$ is prime.

Raphael Tinoco

Numerade Educator

01:55

Problem 50

In a finite group, show that the number of nonidentity elements that satisfy the equation $x^{5}=e$ is a multiple of 5 . If the stipulation that the group be finite is omitted, what can you say about the number of nonidentity elements that satisfy the equation $x^{5}=e$ ?

Kratika Bhadauria

Numerade Educator

05:10

Problem 51

List the six elements of $\mathrm{GL}\left(2, Z_{2}\right) .$ Show that this group is nonAbelian by finding two elements that do not commute. (This exercise is referred to in Chapter $7 .$.)

Ely Crowder

Numerade Educator

12:42

Problem 52

Let $G=\left\{\left[\begin{array}{ll}a & a \\ a & a\end{array}\right] \mid a \in \mathbf{R}, a \neq 0\right\} .$ Show that $G$ is a group under
matrix multiplication. Explain why each element of $G$ has an inverse even though the matrices have 0 determinants. (Compare with Example 10.)

Patrick Vaughan

Numerade Educator

01:58

Problem 53

Suppose that in the definition of a group $G$, the condition that there exists an element $e$ with the property $a e=e a=a$ for all $a$ in $G$ is replaced by $a e=a$ for all $a$ in $G$. Show that $e a=a$ for all $a$ in $G$ (Thus, a one-sided identity is a two-sided identity.)

Varsha Aggarwal

Numerade Educator

05:22

Problem 54

Suppose that in the definition of a group $G$, the condition that for each element $a$ in $G$ there exists an element $b$ in $G$ with the property $a b=b a=e$ is replaced by the condition $a b=e$. Show that $b a=e$. (Thus, a one-sided inverse is a two-sided inverse.)

Yaw Asomani

Numerade Educator