SB Exercises

Exercises 1.5 Exercises

1.

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

$\displaystyle 3x \equiv 2 \pmod{7}$
$\displaystyle 5x + 1 \equiv 13 \pmod{23}$
$\displaystyle 5x + 1 \equiv 13 \pmod{26}$
$\displaystyle 9x \equiv 3 \pmod{5}$
$\displaystyle 5x \equiv 1 \pmod{6}$
$\displaystyle 3x \equiv 1 \pmod{6}$

Hint.

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

2.

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

\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*}
\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*}
\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*}
\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.

3.

Write out Cayley tables for groups formed by the symmetries of a rectangle and for $({\mathbb Z}_4, +)\text{.}$ How many elements are in each group? Are the groups the same? Why or why not?

4.

Describe the symmetries of a rhombus and prove that the set of symmetries forms a group. Give Cayley tables for both the symmetries of a rectangle and the symmetries of a rhombus. Are the symmetries of a rectangle and those of a rhombus the same?

5.

Describe the symmetries of a square and prove that the set of symmetries is a group. Give a Cayley table for the symmetries. How many ways can the vertices of a square be permuted? Is each permutation necessarily a symmetry of the square? The symmetry group of the square is denoted by $D_4\text{.}$

6.

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*}

7.

Let $S = {\mathbb R} \setminus \{ -1 \}$ and define a binary operation on $S$ by $a \ast b = a + b + ab\text{.}$ Prove that $(S, \ast)$ is an abelian group.

8.

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.

9.

Prove that the product of two matrices in $SL_2({\mathbb R})$ has determinant one.

10.

Prove that the set of matrices of the form

\begin{equation*} \begin{pmatrix} 1 & x & y \\ 0 & 1 & z \\ 0 & 0 & 1 \end{pmatrix} \end{equation*}

is a group under matrix multiplication. This group, known as the Heisenberg group, is important in quantum physics. Matrix multiplication in the Heisenberg group is defined by

\begin{equation*} \begin{pmatrix} 1 & x & y \\ 0 & 1 & z \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & x' & y' \\ 0 & 1 & z' \\ 0 & 0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & x+x' & y+y'+xz' \\ 0 & 1 & z+z' \\ 0 & 0 & 1 \end{pmatrix}. \end{equation*}

11.

Prove that $\det(AB) = \det(A) \det(B)$ in $GL_2({\mathbb R})\text{.}$ Use this result to show that the binary operation in the group $GL_2({\mathbb R})$ is closed; that is, if $A$ and $B$ are in $GL_2({\mathbb R})\text{,}$ then $AB \in GL_2({\mathbb R})\text{.}$

12.

Let ${\mathbb Z}_2^n = \{ (a_1, a_2, \ldots, a_n) : a_i \in {\mathbb Z}_2 \}\text{.}$ Define a binary operation on ${\mathbb Z}_2^n$ by

\begin{equation*} (a_1, a_2, \ldots, a_n) + (b_1, b_2, \ldots, b_n) = (a_1 + b_1, a_2 + b_2, \ldots, a_n + b_n). \end{equation*}

Prove that ${\mathbb Z}_2^n$ is a group under this operation. This group is important in algebraic coding theory.

13.

Show that ${\mathbb R}^{\ast} = {\mathbb R} \setminus \{0 \}$ is a group under the operation of multiplication.

14.

Given the groups ${\mathbb R}^{\ast}$ and ${\mathbb Z}\text{,}$ let $G = {\mathbb R}^{\ast} \times {\mathbb Z}\text{.}$ Define a binary operation $\circ$ on $G$ by $(a,m) \circ (b,n) = (ab, m + n)\text{.}$ Show that $G$ is a group under this operation.

15.

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

Hint.

There is a nonabelian group containing six elements.

16.

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.

17.

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

Hint.

The are five different groups of order 8.

18.

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.

19.

Show that

\begin{equation*} 0 + a \equiv a + 0 \equiv a \pmod{ n } \end{equation*}

for all $a \in {\mathbb Z}_n\text{.}$

20.

Prove that there is a multiplicative identity for the integers modulo $n\text{:}$

\begin{equation*} a \cdot 1 \equiv a \pmod{n}. \end{equation*}

21.

For each $a \in {\mathbb Z}_n$ find an element $b \in {\mathbb Z}_n$ such that

\begin{equation*} a + b \equiv b + a \equiv 0 \pmod{ n}. \end{equation*}

22.

Show that addition and multiplication mod $n$ are well defined operations. That is, show that the operations do not depend on the choice of the representative from the equivalence classes mod $n\text{.}$

23.

Show that addition and multiplication mod $n$ are associative operations.

24.

Show that multiplication distributes over addition modulo $n\text{:}$

\begin{equation*} a(b + c) \equiv ab + ac \pmod{n}. \end{equation*}

25.

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*}

26.

Let $U(n)$ be the group of units in ${\mathbb Z}_n\text{.}$ If $n \gt 2\text{,}$ prove that there is an element $k \in U(n)$ such that $k^2 = 1$ and $k \neq 1\text{.}$

27.

Prove that the inverse of $g _1 g_2 \cdots g_n$ is $g_n^{-1} g_{n-1}^{-1} \cdots g_1^{-1}\text{.}$

28.

Prove the remainder of Proposition 1.2.14: if $G$ is a group and $a, b \in G\text{,}$ then the equation $xa = b$ has a unique solution in $G\text{.}$

29.

Prove Theorem 1.2.16.

30.

Prove the right and left cancellation laws for a group $G\text{;}$ that is, show that in the group $G\text{,}$ $ba = ca$ implies $b = c$ and $ab = ac$ implies $b = c$ for elements $a, b, c \in G\text{.}$

31.

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 $ba = ab\text{.}$

32.

Show that if $G$ is a finite group of even order, then there is an $a \in G$ such that $a$ is not the identity and $a^2 = e\text{.}$

33.

Let $G$ be a group and suppose that $(ab)^2 = a^2b^2$ for all $a$ and $b$ in $G\text{.}$ Prove that $G$ is an abelian group.

34.

Find all the subgroups of ${\mathbb Z}_3 \times {\mathbb Z}_3\text{.}$ Use this information to show that ${\mathbb Z}_3 \times {\mathbb Z}_3$ is not the same group as ${\mathbb Z}_9\text{.}$ (See Example 1.3.5 for a short description of the product of groups.)

35.

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

Hint.

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

36.

Compute the subgroups of the symmetry group of a square.

37.

Let $H = \{2^k : k \in {\mathbb Z} \}\text{.}$ Show that $H$ is a subgroup of ${\mathbb Q}^*\text{.}$

38.

Let $n = 0, 1, 2, \ldots$ and $n {\mathbb Z} = \{ nk : k \in {\mathbb Z} \}\text{.}$ Prove that $n {\mathbb Z}$ is a subgroup of ${\mathbb Z}\text{.}$ Show that these subgroups are the only subgroups of $\mathbb{Z}\text{.}$

39.

Let ${\mathbb T} = \{ z \in {\mathbb C}^* : |z| =1 \}\text{.}$ Prove that ${\mathbb T}$ is a subgroup of ${\mathbb C}^*\text{.}$

40.

\begin{equation*} \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix} \end{equation*}

where $\theta \in {\mathbb R}\text{.}$ Prove that $G$ is a subgroup of $SL_2({\mathbb R})\text{.}$

41.

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, $(a + b \sqrt{2}\, )^{-1} = a/(a^2 - 2b^2) - b\sqrt{2}/(a^2 - 2 b^2)\text{.}$

42.

Let $G$ be the group of $2 \times 2$ matrices under addition and

\begin{equation*} H = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} : a + d = 0 \right\}. \end{equation*}

Prove that $H$ is a subgroup of $G\text{.}$

43.

Prove or disprove: $SL_2( {\mathbb Z} )\text{,}$ the set of $2 \times 2$ matrices with integer entries and determinant one, is a subgroup of $SL_2( {\mathbb R} )\text{.}$

44.

List the subgroups of the quaternion group, $Q_8\text{.}$

45.

Prove that the intersection of two subgroups of a group $G$ is also a subgroup of $G\text{.}$

46.

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 $S_3\text{.}$

47.

Prove or disprove: If $H$ and $K$ are subgroups of a group $G\text{,}$ then $H K = \{hk : h \in H \text{ and } k \in K \}$ is a subgroup of $G\text{.}$ What if $G$ is abelian?

48.

Let $G$ be a group and $g \in G\text{.}$ Show that

\begin{equation*} Z(G) = \{ x \in G : gx = xg \text{ for all } g \in G \} \end{equation*}

is a subgroup of $G\text{.}$ This subgroup is called the center of $G\text{.}$

49.

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 $ab = a^3 b a = ba\text{.}$

50.

Give an example of an infinite group in which every nontrivial subgroup is infinite.

51.

If $xy = x^{-1} y^{-1}$ for all $x$ and $y$ in $G\text{,}$ prove that $G$ must be abelian.

52.

Prove or disprove: Every proper subgroup of an nonabelian group is nonabelian.

53.

Let $H$ be a subgroup of $G$ and

\begin{equation*} C(H) = \{ g \in G : gh = hg \text{ for all } h \in H \}. \end{equation*}

Prove $C(H)$ is a subgroup of $G\text{.}$ This subgroup is called the centralizer of $H$ in $G\text{.}$

54.

Let $H$ be a subgroup of $G\text{.}$ If $g \in G\text{,}$ show that $gHg^{-1} = \{g^{-1}hg : h\in H\}$ is also a subgroup of $G\text{.}$

Exercise Group.

In each group, how many solutions are there to $x^2=e\text{?}$

55.

$C_n\text{,}$ $n$ odd.

Answer.

$1$

56.

$C_n\text{,}$ $n$ even.

Answer.

$2$

57.

$D_n\text{,}$ $n$ odd.

Answer.

$n$

58.

$D_n\text{,}$ $n$ even.

Answer.

$n+1$

59.

This is an odd-numbered exercise with tasks.

(a)

What is $1+1\text{?}$

Answer.

$2$

(b)

This task has subtasks.

(i)

What is $3+3\text{?}$

Answer.

$6$

(ii)

What is $5+5\text{?}$

Answer.

$10$

60.

This is an even-numbered exercise with tasks.

(a)

What is $2+2\text{?}$

Answer.

$4$

(b)

This task has subtasks.

(i)

What is $4+4\text{?}$

Answer.

$8$

(ii)

What is $6+6\text{?}$

Answer.

$12$

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