
# Exercises1.3Exercises

###### Exercise1

Suppose that

\begin{align*} A & = \{ x : x \in \mathbb N \text{ and } x \text{ is even} \},\\ B & = \{x : x \in \mathbb N \text{ and } x \text{ is prime}\},\\ C & = \{ x : x \in \mathbb N \text{ and } x \text{ is a multiple of 5}\}. \end{align*}

Describe each of the following sets.

1. $A \cap B$

2. $B \cap C$

3. $A \cup B$

4. $A \cap (B \cup C)$

Hint

(a) $A \cap B = \{ 2 \}\text{;}$ (b) $B \cap C = \{ 5 \}\text{.}$

###### Exercise2

If $A = \{ a, b, c \}\text{,}$ $B = \{ 1, 2, 3 \}\text{,}$ $C = \{ x \}\text{,}$ and $D = \emptyset\text{,}$ list all of the elements in each of the following sets.

1. $A \times B$

2. $B \times A$

3. $A \times B \times C$

4. $A \times D$

Hint

(a) $A \times B = \{ (a,1), (a,2), (a,3), (b,1), (b,2), (b,3), (c,1), (c,2), (c,3) \}\text{;}$ (d) $A \times D = \emptyset\text{.}$

###### Exercise6

Prove $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)\text{.}$

Hint

If $x \in A \cup (B \cap C)\text{,}$ then either $x \in A$ or $x \in B \cap C\text{.}$ Thus, $x \in A \cup B$ and $A \cup C\text{.}$ Hence, $x \in (A \cup B) \cap (A \cup C)\text{.}$ Therefore, $A \cup (B \cap C) \subset (A \cup B) \cap (A \cup C)\text{.}$ Conversely, if $x \in (A \cup B) \cap (A \cup C)\text{,}$ then $x \in A \cup B$ and $A \cup C\text{.}$ Thus, $x \in A$ or $x$ is in both $B$ and $C\text{.}$ So $x \in A \cup (B \cap C)$ and therefore $(A \cup B) \cap (A \cup C) \subset A \cup (B \cap C)\text{.}$ Hence, $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)\text{.}$

###### Exercise10

Prove $A \cup B = (A \cap B) \cup (A \setminus B) \cup (B \setminus A)\text{.}$

Hint

$(A \cap B) \cup (A \setminus B) \cup (B \setminus A) = (A \cap B) \cup (A \cap B') \cup (B \cap A') = [A \cap (B \cup B')] \cup (B \cap A') = A \cup (B \cap A') = (A \cup B) \cap (A \cup A') = A \cup B\text{.}$

###### Exercise14

Prove $A \setminus (B \cup C) = (A \setminus B) \cap (A \setminus C)\text{.}$

Hint

$A \setminus (B \cup C) = A \cap (B \cup C)' = (A \cap A) \cap (B' \cap C') = (A \cap B') \cap (A \cap C') = (A \setminus B) \cap (A \setminus C)\text{.}$

###### Exercise17

Which of the following relations $f: {\mathbb Q} \rightarrow {\mathbb Q}$ define a mapping? In each case, supply a reason why $f$ is or is not a mapping.

1. $\displaystyle f(p/q) = \frac{p+ 1}{p - 2}$

2. $\displaystyle f(p/q) = \frac{3p}{3q}$

3. $\displaystyle f(p/q) = \frac{p+q}{q^2}$

4. $\displaystyle f(p/q) = \frac{3 p^2}{7 q^2} - \frac{p}{q}$

Hint

(a) Not a map since $f(2/3)$ is undefined; (b) this is a map; (c) not a map, since $f(1/2) = 3/4$ but $f(2/4)=3/8\text{;}$ (d) this is a map.

###### Exercise18

Determine which of the following functions are one-to-one and which are onto. If the function is not onto, determine its range.

1. $f: {\mathbb R} \rightarrow {\mathbb R}$ defined by $f(x) = e^x$

2. $f: {\mathbb Z} \rightarrow {\mathbb Z}$ defined by $f(n) = n^2 + 3$

3. $f: {\mathbb R} \rightarrow {\mathbb R}$ defined by $f(x) = \sin x$

4. $f: {\mathbb Z} \rightarrow {\mathbb Z}$ defined by $f(x) = x^2$

Hint

(a) $f$ is one-to-one but not onto. $f({\mathbb R} ) = \{ x \in {\mathbb R} : x \gt 0 \}\text{.}$ (c) $f$ is neither one-to-one nor onto. $f(\mathbb R) = \{ x : -1 \leq x \leq 1 \}\text{.}$

###### Exercise20
1. Define a function $f: {\mathbb N} \rightarrow {\mathbb N}$ that is one-to-one but not onto.

2. Define a function $f: {\mathbb N} \rightarrow {\mathbb N}$ that is onto but not one-to-one.

Hint

(a) $f(n) = n + 1\text{.}$

###### Exercise22

Let $f : A \rightarrow B$ and $g : B \rightarrow C$ be maps.

1. If $f$ and $g$ are both one-to-one functions, show that $g \circ f$ is one-to-one.

2. If $g \circ f$ is onto, show that $g$ is onto.

3. If $g \circ f$ is one-to-one, show that $f$ is one-to-one.

4. If $g \circ f$ is one-to-one and $f$ is onto, show that $g$ is one-to-one.

5. If $g \circ f$ is onto and $g$ is one-to-one, show that $f$ is onto.

Hint

(a) Let $x, y \in A\text{.}$ Then $g(f(x)) = (g \circ f)(x) = (g \circ f)(y) = g(f(y))\text{.}$ Thus, $f(x) = f(y)$ and $x = y\text{,}$ so $g \circ f$ is one-to-one. (b) Let $c \in C\text{,}$ then $c = (g \circ f)(x) = g(f(x))$ for some $x \in A\text{.}$ Since $f(x) \in B\text{,}$ $g$ is onto.

###### Exercise23

Define a function on the real numbers by

\begin{equation*} f(x) = \frac{x + 1}{x - 1}. \end{equation*}

What are the domain and range of $f\text{?}$ What is the inverse of $f\text{?}$ Compute $f \circ f^{-1}$ and $f^{-1} \circ f\text{.}$

Hint

$f^{-1}(x) = (x+1)/(x-1)\text{.}$

###### Exercise24

Let $f: X \rightarrow Y$ be a map with $A_1, A_2 \subset X$ and $B_1, B_2 \subset Y\text{.}$

1. Prove $f( A_1 \cup A_2 ) = f( A_1) \cup f( A_2 )\text{.}$

2. Prove $f( A_1 \cap A_2 ) \subset f( A_1) \cap f( A_2 )\text{.}$ Give an example in which equality fails.

3. Prove $f^{-1}( B_1 \cup B_2 ) = f^{-1}( B_1) \cup f^{-1}(B_2 )\text{,}$ where

\begin{equation*} f^{-1}(B) = \{ x \in X : f(x) \in B \}. \end{equation*}
4. Prove $f^{-1}( B_1 \cap B_2 ) = f^{-1}( B_1) \cap f^{-1}( B_2 )\text{.}$

5. Prove $f^{-1}( Y \setminus B_1 ) = X \setminus f^{-1}( B_1)\text{.}$

Hint

(a) Let $y \in f(A_1 \cup A_2)\text{.}$ Then there exists an $x \in A_1 \cup A_2$ such that $f(x) = y\text{.}$ Hence, $y \in f(A_1)$ or $f(A_2) \text{.}$ Therefore, $y \in f(A_1) \cup f(A_2)\text{.}$ Consequently, $f(A_1 \cup A_2) \subset f(A_1) \cup f(A_2)\text{.}$ Conversely, if $y \in f(A_1) \cup f(A_2)\text{,}$ then $y \in f(A_1)$ or $f(A_2)\text{.}$ Hence, there exists an $x \in A_1$ or there exists an $x \in A_2$ such that $f(x) = y\text{.}$ Thus, there exists an $x \in A_1 \cup A_2$ such that $f(x) = y\text{.}$ Therefore, $f(A_1) \cup f(A_2) \subset f(A_1 \cup A_2)\text{,}$ and $f(A_1 \cup A_2) = f(A_1) \cup f(A_2)\text{.}$

###### Exercise25

Determine whether or not the following relations are equivalence relations on the given set. If the relation is an equivalence relation, describe the partition given by it. If the relation is not an equivalence relation, state why it fails to be one.

1. $x \sim y$ in ${\mathbb R}$ if $x \geq y$

2. $m \sim n$ in ${\mathbb Z}$ if $mn > 0$

3. $x \sim y$ in ${\mathbb R}$ if $|x - y| \leq 4$

4. $m \sim n$ in ${\mathbb Z}$ if $m \equiv n \pmod{6}$

Hint

(a) NThe relation fails to be symmetric. (b) The relation is not reflexive, since 0 is not equivalent to itself. (c) The relation is not transitive.

###### Exercise28

Find the error in the following argument by providing a counterexample. “The reflexive property is redundant in the axioms for an equivalence relation. If $x \sim y\text{,}$ then $y \sim x$ by the symmetric property. Using the transitive property, we can deduce that $x \sim x\text{.}$”

Hint

Let $X = {\mathbb N} \cup \{ \sqrt{2}\, \}$ and define $x \sim y$ if $x + y \in {\mathbb N}\text{.}$

# Exercises2.3Exercises

###### Exercise1

Prove that

\begin{equation*} 1^2 + 2^2 + \cdots + n^2 = \frac{n(n + 1)(2n + 1)}{6} \end{equation*}

for $n \in {\mathbb N}\text{.}$

Hint

The base case, $S(1): [1(1 + 1)(2(1) + 1)]/6 = 1 = 1^2$ is true. Assume that $S(k): 1^2 + 2^2 + \cdots + k^2 = [k(k + 1)(2k + 1)]/6$ is true. Then

\begin{align*} 1^2 + 2^2 + \cdots + k^2 + (k + 1)^2 & = [k(k + 1)(2k + 1)]/6 + (k + 1)^2\\ & = [(k + 1)((k + 1) + 1)(2(k + 1) + 1)]/6, \end{align*}

and so $S(k + 1)$ is true. Thus, $S(n)$ is true for all positive integers $n\text{.}$

###### Exercise3

Prove that $n! \gt 2^n$ for $n \geq 4\text{.}$

Hint

The base case, $S(4): 4! = 24 \gt 16 =2^4$ is true. Assume $S(k): k! \gt 2^k$ is true. Then $(k + 1)! = k! (k + 1) \gt 2^k \cdot 2 = 2^{k + 1}\text{,}$ so $S(k + 1)$ is true. Thus, $S(n)$ is true for all positive integers $n\text{.}$

###### Exercise8

Prove the Leibniz rule for $f^{(n)} (x)\text{,}$ where $f^{(n)}$ is the $n$th derivative of $f\text{;}$ that is, show that

\begin{equation*} (fg)^{(n)}(x) = \sum_{k = 0}^{n} \binom{n}{k} f^{(k)}(x) g^{(n - k)}(x). \end{equation*}
Hint

Follow the proof in Example Example 2.1.4.

###### Exercise11

If $x$ is a nonnegative real number, then show that $(1 + x)^n - 1 \geq nx$ for $n = 0, 1, 2, \ldots\text{.}$

Hint

The base case, $S(0): (1 + x)^0 - 1 = 0 \geq 0 = 0 \cdot x$ is true. Assume $S(k): (1 + x)^k -1 \geq kx$ is true. Then

\begin{align*} (1 + x)^{k + 1} - 1 & = (1 + x)(1 + x)^k -1\\ & = (1 + x)^k + x(1 + x)^k - 1\\ & \geq kx + x(1 + x)^k\\ & \geq kx + x\\ & = (k + 1)x, \end{align*}

so $S(k + 1)$ is true. Therefore, $S(n)$ is true for all positive integers $n\text{.}$

###### Exercise17Fibonacci Numbers

The Fibonacci numbers are

\begin{equation*} 1, 1, 2, 3, 5, 8, 13, 21, \ldots. \end{equation*}

We can define them inductively by $f_1 = 1\text{,}$ $f_2 = 1\text{,}$ and $f_{n + 2} = f_{n + 1} + f_n$ for $n \in {\mathbb N}\text{.}$

1. Prove that $f_n \lt 2^n\text{.}$

2. Prove that $f_{n + 1} f_{n - 1} = f^2_n + (-1)^n\text{,}$ $n \geq 2\text{.}$

3. Prove that $f_n = [(1 + \sqrt{5}\, )^n - (1 - \sqrt{5}\, )^n]/ 2^n \sqrt{5}\text{.}$

4. Show that $\lim_{n \rightarrow \infty} f_n / f_{n + 1} = (\sqrt{5} - 1)/2\text{.}$

5. Prove that $f_n$ and $f_{n + 1}$ are relatively prime.

Hint

For (a) and (b) use mathematical induction. (c) Show that $f_1 = 1\text{,}$ $f_2 = 1\text{,}$ and $f_{n + 2} = f_{n + 1} + f_n\text{.}$ (d) Use part (c). (e) Use part (b) and Exercise Exercise 2.3.16.

###### Exercise19

Let $x, y \in {\mathbb N}$ be relatively prime. If $xy$ is a perfect square, prove that $x$ and $y$ must both be perfect squares.

Hint

Use the Fundamental Theorem of Arithmetic.

###### Exercise23

Define the least common multiple of two nonzero integers $a$ and $b\text{,}$ denoted by $\lcm(a,b)\text{,}$ to be the nonnegative integer $m$ such that both $a$ and $b$ divide $m\text{,}$ and if $a$ and $b$ divide any other integer $n\text{,}$ then $m$ also divides $n\text{.}$ Prove that any two integers $a$ and $b$ have a unique least common multiple.

Hint

Let $S = \{s \in {\mathbb N} : a \mid s\text{,}$ $b \mid s \}\text{.}$ Then $S \neq \emptyset\text{,}$ since $|ab| \in S\text{.}$ By the Principle of Well-Ordering, $S$ contains a least element $m\text{.}$ To show uniqueness, suppose that $a \mid n$ and $b \mid n$ for some $n \in {\mathbb N}\text{.}$ By the division algorithm, there exist unique integers $q$ and $r$ such that $n = mq + r\text{,}$ where $0 \leq r \lt m\text{.}$ Since $a$ and $b$ divide both $m\text{,}$ and $n\text{,}$ it must be the case that $a$ and $b$ both divide $r\text{.}$ Thus, $r = 0$ by the minimality of $m\text{.}$ Therefore, $m \mid n\text{.}$

###### Exercise27

Let $a, b, c \in {\mathbb Z}\text{.}$ Prove that if $\gcd(a,b) = 1$ and $a \mid bc\text{,}$ then $a \mid c\text{.}$

Hint

Since $\gcd(a,b) = 1\text{,}$ there exist integers $r$ and $s$ such that $ar + bs = 1\text{.}$ Thus, $acr + bcs = c\text{.}$ Since $a$ divides both $bc$ and itself, $a$ must divide $c\text{.}$

###### Exercise29

Prove that there are an infinite number of primes of the form $6n + 5\text{.}$

Hint

Every prime must be of the form 2, 3, $6n + 1\text{,}$ or $6n + 5\text{.}$ Suppose there are only finitely many primes of the form $6k + 5\text{.}$

# Exercises3.4Exercises

###### Exercise1

Find all $x \in {\mathbb Z}$ satisfying each of the following equations.

1. $3x \equiv 2 \pmod{7}$

2. $5x + 1 \equiv 13 \pmod{23}$

3. $5x + 1 \equiv 13 \pmod{26}$

4. $9x \equiv 3 \pmod{5}$

5. $5x \equiv 1 \pmod{6}$

6. $3x \equiv 1 \pmod{6}$

Hint

(a) $3 + 7 \mathbb Z = \{ \ldots, -4, 3, 10, \ldots \}\text{;}$ (c) $18 + 26 \mathbb Z\text{;}$ (e) $5 + 6 \mathbb Z\text{.}$

###### Exercise2

Which of the following multiplication tables defined on the set $G = \{ a, b, c, d \}$ form a group? Support your answer in each case.

1. \begin{equation*} \begin{array}{c|cccc} \circ & a & b & c & d \\ \hline a & a & c & d & a \\ b & b & b & c & d \\ c & c & d & a & b \\ d & d & a & b & c \end{array} \end{equation*}
2. \begin{equation*} \begin{array}{c|cccc} \circ & a & b & c & d \\ \hline a & a & b & c & d \\ b & b & a & d & c \\ c & c & d & a & b \\ d & d & c & b & a \end{array} \end{equation*}
3. \begin{equation*} \begin{array}{c|cccc} \circ & a & b & c & d \\ \hline a & a & b & c & d \\ b & b & c & d & a \\ c & c & d & a & b \\ d & d & a & b & c \end{array} \end{equation*}
4. \begin{equation*} \begin{array}{c|cccc} \circ & a & b & c & d \\ \hline a & a & b & c & d \\ b & b & a & c & d \\ c & c & b & a & d \\ d & d & d & b & c \end{array} \end{equation*}
Hint

(a) Not a group; (c) a group.

###### Exercise6

Give a multiplication table for the group $U(12)\text{.}$

Hint

\begin{equation*} \begin{array}{c|cccc} \cdot & 1 & 5 & 7 & 11 \\ \hline 1 & 1 & 5 & 7 & 11 \\ 5 & 5 & 1 & 11 & 7 \\ 7 & 7 & 11 & 1 & 5 \\ 11 & 11 & 7 & 5 & 1 \end{array} \end{equation*}
###### Exercise8

Give an example of two elements $A$ and $B$ in $GL_2({\mathbb R})$ with $AB \neq BA\text{.}$

Hint

Pick two matrices. Almost any pair will work.

###### Exercise15

Prove or disprove that every group containing six elements is abelian.

Hint

There is a nonabelian group containing six elements.

###### Exercise16

Give a specific example of some group $G$ and elements $g, h \in G$ where $(gh)^n \neq g^nh^n\text{.}$

Hint

Look at the symmetry group of an equilateral triangle or a square.

###### Exercise17

Give an example of three different groups with eight elements. Why are the groups different?

Hint

The are five different groups of order 8.

###### Exercise18

Show that there are $n!$ permutations of a set containing $n$ items.

Hint

Let

\begin{equation*} \sigma = \begin{pmatrix} 1 & 2 & \cdots & n \\ a_1 & a_2 & \cdots & a_n \end{pmatrix} \end{equation*}

be in $S_n\text{.}$ All of the $a_i$s must be distinct. There are $n$ ways to choose $a_1\text{,}$ $n-1$ ways to choose $a_2\text{,}$ $\ldots\text{,}$ 2 ways to choose $a_{n - 1}\text{,}$ and only one way to choose $a_n\text{.}$ Therefore, we can form $\sigma$ in $n(n - 1) \cdots 2 \cdot 1 = n!$ ways.

###### Exercise25

Let $a$ and $b$ be elements in a group $G\text{.}$ Prove that $ab^na^{-1} = (aba^{-1})^n$ for $n \in \mathbb Z\text{.}$

Hint

\begin{align*} (aba^{-1})^n & = (aba^{-1})(aba^{-1}) \cdots (aba^{-1})\\ & = ab(aa^{-1})b(aa^{-1})b \cdots b(aa^{-1})ba^{-1}\\ & = ab^na^{-1}. \end{align*}
###### Exercise31

Show that if $a^2 = e$ for all elements $a$ in a group $G\text{,}$ then $G$ must be abelian.

Hint

Since $abab = (ab)^2 = e = a^2 b^2 = aabb\text{,}$ we know that $ba = ab\text{.}$

###### Exercise35

Find all the subgroups of the symmetry group of an equilateral triangle.

Hint

$H_1 = \{ id \}\text{,}$ $H_2 = \{ id, \rho_1, \rho_2 \}\text{,}$ $H_3 = \{ id, \mu_1 \}\text{,}$ $H_4 = \{ id, \mu_2 \}\text{,}$ $H_5 = \{ id, \mu_3 \}\text{,}$ $S_3\text{.}$

###### Exercise41

Prove that

\begin{equation*} G = \{ a + b \sqrt{2} : a, b \in {\mathbb Q} \text{ and } a \text{ and } b \text{ are not both zero} \} \end{equation*}

is a subgroup of ${\mathbb R}^{\ast}$ under the group operation of multiplication.

Hint

The identity of $G$ is $1 = 1 + 0 \sqrt{2}\text{.}$ Since $(a + b \sqrt{2}\, )(c + d \sqrt{2}\, ) = (ac + 2bd) + (ad + bc)\sqrt{2}\text{,}$ $G$ is closed under multiplication. Finally, $(a + b \sqrt{2}\, )^{-1} = a/(a^2 - 2b^2) - b\sqrt{2}/(a^2 - 2 b^2)\text{.}$

###### Exercise46

Prove or disprove: If $H$ and $K$ are subgroups of a group $G\text{,}$ then $H \cup K$ is a subgroup of $G\text{.}$

Hint

Look at $S_3\text{.}$

###### Exercise49

Let $a$ and $b$ be elements of a group $G\text{.}$ If $a^4b = ba$ and $a^3 = e\text{,}$ prove that $ab = ba\text{.}$

Hint

Since $a^4b = ba\text{,}$ it must be the case that $b = a^6 b = a^2 b a\text{,}$ and we can conclude that $ab = a^3 b a = ba\text{.}$

# Exercises4.5Exercises

###### Exercise1

Prove or disprove each of the following statements.

1. All of the generators of ${\mathbb Z}_{60}$ are prime.

2. $U(8)$ is cyclic.

3. ${\mathbb Q}$ is cyclic.

4. If every proper subgroup of a group $G$ is cyclic, then $G$ is a cyclic group.

5. A group with a finite number of subgroups is finite.

Hint

(a) False; (c) false; (e) true.

###### Exercise2

Find the order of each of the following elements.

1. $5 \in {\mathbb Z}_{12}$

2. $\sqrt{3} \in {\mathbb R}$

3. $\sqrt{3} \in {\mathbb R}^\ast$

4. $-i \in {\mathbb C}^\ast$

5. 72 in ${\mathbb Z}_{240}$

6. 312 in ${\mathbb Z}_{471}$

Hint

(a) 12; (c) infinite; (e) 10.

###### Exercise3

List all of the elements in each of the following subgroups.

1. The subgroup of ${\mathbb Z}$ generated by 7

2. The subgroup of ${\mathbb Z}_{24}$ generated by 15

3. All subgroups of ${\mathbb Z}_{12}$

4. All subgroups of ${\mathbb Z}_{60}$

5. All subgroups of ${\mathbb Z}_{13}$

6. All subgroups of ${\mathbb Z}_{48}$

7. The subgroup generated by 3 in $U(20)$

8. The subgroup generated by 5 in $U(18)$

9. The subgroup of ${\mathbb R}^\ast$ generated by 7

10. The subgroup of ${\mathbb C}^\ast$ generated by $i$ where $i^2 = -1$

11. The subgroup of ${\mathbb C}^\ast$ generated by $2i$

12. The subgroup of ${\mathbb C}^\ast$ generated by $(1 + i) / \sqrt{2}$

13. The subgroup of ${\mathbb C}^\ast$ generated by $(1 + \sqrt{3}\, i) / 2$

Hint

(a) $7 {\mathbb Z} = \{ \ldots, -7, 0, 7, 14, \ldots \}\text{;}$ (b) $\{ 0, 3, 6, 9, 12, 15, 18, 21 \}\text{;}$ (c) $\{ 0 \}\text{,}$ $\{ 0, 6 \}\text{,}$ $\{ 0, 4, 8 \}\text{,}$ $\{ 0, 3, 6, 9 \}\text{,}$ $\{ 0, 2, 4, 6, 8, 10 \}\text{;}$ (g) $\{ 1, 3, 7, 9 \}\text{;}$ (j) $\{ 1, -1, i, -i \}\text{.}$

###### Exercise4

Find the subgroups of $GL_2( {\mathbb R })$ generated by each of the following matrices.

1. $\displaystyle \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$

2. $\displaystyle \begin{pmatrix} 0 & 1/3 \\ 3 & 0 \end{pmatrix}$

3. $\displaystyle \begin{pmatrix} 1 & -1 \\ 1 & 0 \end{pmatrix}$

4. $\displaystyle \begin{pmatrix} 1 & -1 \\ 0 & 1 \end{pmatrix}$

5. $\displaystyle \begin{pmatrix} 1 & -1 \\ -1 & 0 \end{pmatrix}$

6. $\displaystyle \begin{pmatrix} \sqrt{3}/ 2 & 1/2 \\ -1/2 & \sqrt{3}/2 \end{pmatrix}$

Hint

(a)

\begin{equation*} \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}, \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}. \end{equation*}

(c)

\begin{equation*} \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \begin{pmatrix} 1 & -1 \\ 1 & 0 \end{pmatrix}, \begin{pmatrix} -1 & 1 \\ -1 & 0 \end{pmatrix}, \\ \begin{pmatrix} 0 & 1 \\ -1 & 1 \end{pmatrix}, \begin{pmatrix} 0 & -1 \\ 1 & -1 \end{pmatrix}, \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}. \end{equation*}
###### Exercise10

Find all elements of finite order in each of the following groups. Here the “$\ast$” indicates the set with zero removed.

1. ${\mathbb Z}$

2. ${\mathbb Q}^\ast$

3. ${\mathbb R}^\ast$

Hint

(a) $0, 1, -1\text{;}$ (b) $1, -1$

###### Exercise11

If $a^{24} =e$ in a group $G\text{,}$ what are the possible orders of $a\text{?}$

Hint

1, 2, 3, 4, 6, 8, 12, 24.

###### Exercise15

Evaluate each of the following.

1. $(3-2i)+ (5i-6)$

2. $(4-5i)-\overline{(4i -4)}$

3. $(5-4i)(7+2i)$

4. $(9-i) \overline{(9-i)}$

5. $i^{45}$

6. $(1+i)+\overline{(1+i)}$

Hint

(a) $-3 + 3i\text{;}$ (c) $43- 18i\text{;}$ (e) $i$

###### Exercise16

Convert the following complex numbers to the form $a + bi\text{.}$

1. $2 \cis(\pi / 6 )$

2. $5 \cis(9\pi/4)$

3. $3 \cis(\pi)$

4. $\cis(7\pi/4) /2$

Hint

(a) $\sqrt{3} + i\text{;}$ (c) $-3\text{.}$

###### Exercise17

Change the following complex numbers to polar representation.

1. $1-i$

2. $-5$

3. $2+2i$

4. $\sqrt{3} + i$

5. $-3i$

6. $2i + 2 \sqrt{3}$

Hint

(a) $\sqrt{2} \cis( 7 \pi /4)\text{;}$ (c) $2 \sqrt{2} \cis( \pi /4)\text{;}$ (e) $3 \cis(3 \pi/2)\text{.}$

###### Exercise18

Calculate each of the following expressions.

1. $(1+i)^{-1}$

2. $(1 - i)^{6}$

3. $(\sqrt{3} + i)^{5}$

4. $(-i)^{10}$

5. $((1-i)/2)^{4}$

6. $(-\sqrt{2} - \sqrt{2}\, i)^{12}$

7. $(-2 + 2i)^{-5}$

Hint

(a) $(1 - i)/2\text{;}$ (c) $16(i - \sqrt{3}\, )\text{;}$ (e) $-1/4\text{.}$

###### Exercise22

Calculate each of the following.

1. $292^{3171} \pmod{ 582}$

2. $2557^{ 341} \pmod{ 5681}$

3. $2071^{ 9521} \pmod{ 4724}$

4. $971^{ 321} \pmod{ 765}$

Hint

(a) 292; (c) 1523.

###### Exercise27

If $g$ and $h$ have orders 15 and 16 respectively in a group $G\text{,}$ what is the order of $\langle g \rangle \cap \langle h \rangle \text{?}$

Hint

$|\langle g \rangle \cap \langle h \rangle| = 1\text{.}$

###### Exercise31

Let $G$ be an abelian group. Show that the elements of finite order in $G$ form a subgroup. This subgroup is called the torsion subgroup of $G\text{.}$

Hint

The identity element in any group has finite order. Let $g, h \in G$ have orders $m$ and $n\text{,}$ respectively. Since $(g^{-1})^m = e$ and $(gh)^{mn} = e\text{,}$ the elements of finite order in $G$ form a subgroup of $G\text{.}$

###### Exercise37

Prove that if $G$ has no proper nontrivial subgroups, then $G$ is a cyclic group.

Hint

If $g$ is an element distinct from the identity in $G\text{,}$ $g$ must generate $G\text{;}$ otherwise, $\langle g \rangle$ is a nontrivial proper subgroup of $G\text{.}$