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Section3.4Exercises

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
1

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) $5 + 6 \\mathbb Z\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
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.

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

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

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
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?

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
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?

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
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 $D_4\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
6

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

Hint
\n\\begin{equation*}\n\\begin{array}{c|cccc}\n\\cdot & 1 & 5 & 7 & 11 \\\\\n\\hline\n1 & 1 & 5 & 7 & 11 \\\\ 5 & 5 & 1 & 11 & 7 \\\\\n7 & 7 & 11 & 1 & 5 \\\\ 11 & 11 & 7 & 5 & 1\n\\end{array}\n\\end{equation*}\n
"]}, {"cell_type":"markdown", "metadata":{}, "source":["
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.

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
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.

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
9

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

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
"]}, {"cell_type":"markdown", "metadata":{}, "source":["
10

Prove that the set of matrices of the form

\n\\begin{equation*}\n\\begin{pmatrix}\n1 & x & y \\\\\n0 & 1 & z \\\\\n0 & 0 & 1\n\\end{pmatrix}\n\\end{equation*}\n

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

\n\\begin{equation*}\n\\begin{pmatrix}\n1 & x & y \\\\\n0 & 1 & z \\\\\n0 & 0 & 1\n\\end{pmatrix}\n\\begin{pmatrix}\n1 & x' & y' \\\\\n0 & 1 & z' \\\\\n0 & 0 & 1\n\\end{pmatrix}\n=\n\\begin{pmatrix}\n1 & x+x' & y+y'+xz' \\\\\n0 & 1 & z+z' \\\\\n0 & 0 & 1\n\\end{pmatrix}.\n\\end{equation*}\n
"]}, {"cell_type":"markdown", "metadata":{}, "source":["
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 $AB \\in GL_2({\\mathbb R})\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
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

\n\\begin{equation*}\n(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).\n\\end{equation*}\n

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

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
13

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

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
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.

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
15

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

Hint

There is a nonabelian group containing six elements.

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
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.

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
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.

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
18

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

Hint

Let

\n\\begin{equation*}\n\\sigma =\n\\begin{pmatrix}\n1 & 2 & \\cdots & n \\\\ a_1 & a_2 & \\cdots & a_n\n\\end{pmatrix}\n\\end{equation*}\n

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.

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
19

Show that

\n\\begin{equation*}\n0 + a \\equiv a + 0 \\equiv a \\pmod{ n }\n\\end{equation*}\n

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

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
20

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

\n\\begin{equation*}\na \\cdot 1 \\equiv a \\pmod{n}.\n\\end{equation*}\n
"]}, {"cell_type":"markdown", "metadata":{}, "source":["
21

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

\n\\begin{equation*}\na + b \\equiv b + a \\equiv 0 \\pmod{ n}.\n\\end{equation*}\n
"]}, {"cell_type":"markdown", "metadata":{}, "source":["
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 $n\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
23

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

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
24

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

\n\\begin{equation*}\na(b + c) \\equiv ab + ac \\pmod{n}.\n\\end{equation*}\n
"]}, {"cell_type":"markdown", "metadata":{}, "source":["
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
\n\\begin{align*}\n(aba^{-1})^n & = (aba^{-1})(aba^{-1}) \\cdots (aba^{-1})\\\\\n& = ab(aa^{-1})b(aa^{-1})b \\cdots b(aa^{-1})ba^{-1}\\\\\n& = ab^na^{-1}.\n\\end{align*}\n
"]}, {"cell_type":"markdown", "metadata":{}, "source":["
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 $k \\neq 1\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
27

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

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
28

Prove the remainder of Proposition 3.21: if $G$ is a group and $a, b \\in G\\text{,}$ then the equation $xa = b$ has a unique solution in $G\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
29

Prove Theorem 3.23.

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
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 $a, b, c \\in G\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
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 $ba = ab\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
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 $a^2 = e\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
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.

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
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 3.28 for a short description of the product of groups.)

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
35

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

Hint

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

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
36

Compute the subgroups of the symmetry group of a square.

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
37

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

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
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 $\\mathbb{Z}\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
39

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

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
40

Let $G$ consist of the $2 \\times 2$ matrices of the form

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

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

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
41

Prove that

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

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

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
42

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

\n\\begin{equation*}\nH =\n\\left\\{\n\\begin{pmatrix}\na & b \\\\\nc & d\n\\end{pmatrix} :\na + d = 0\n\\right\\}.\n\\end{equation*}\n

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

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
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 $SL_2( {\\mathbb R} )\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
44

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

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
45

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

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
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 $S_3\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
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?

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
48

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

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

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

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
49

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

Hint

$b a = a^4 b = a^3 a b = ab$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
50

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

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
"]}, {"cell_type":"markdown", "metadata":{}, "source":["
51

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

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
52

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

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
"]}, {"cell_type":"markdown", "metadata":{}, "source":["
53

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

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

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

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
54

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

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