<exercises xml:id="exercises-groups" xml:base="exercises/groups.xml">
<title>Exercises</title>
<exercise number="1">
<statement>
<p>
Find all <m>x \in {\mathbb Z}</m> satisfying each of the following equations.
</p>
<ol cols="2">
<li>
<p>
<m>3x \equiv 2 \pmod{7}</m>
</p>
</li>
<li>
<p>
<m>5x + 1 \equiv 13 \pmod{23}</m>
</p>
</li>
<li>
<p>
<m>5x + 1 \equiv 13 \pmod{26}</m>
</p>
</li>
<li>
<p>
<m>9x \equiv 3 \pmod{5}</m>
</p>
</li>
<li>
<p>
<m>5x \equiv 1 \pmod{6}</m>
</p>
</li>
<li>
<p>
<m>3x \equiv 1 \pmod{6}</m>
</p>
</li>
</ol>
</statement>
<hint>
<p>
(a) <m>3 + 7 \mathbb Z = \{ \ldots, -4, 3, 10, \ldots \}</m>; (c)
<m>18 + 26 \mathbb Z</m>; (e) <m>5 + 6 \mathbb Z</m>.
</p>
</hint>
</exercise>
<exercise number="2">
<statement>
<p>
Which of the following multiplication tables defined on the set <m>G = \{ a, b, c, d \}</m> form a group?
Support your answer in each case.
</p>
<ol cols="2">
<li>
<p>
<me>
\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}
</me>
</p>
</li>
<li>
<p>
<me>
\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}
</me>
</p>
</li>
<li>
<p>
<me>
\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}
</me>
</p>
</li>
<li>
<p>
<me>
\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}
</me>
</p>
</li>
</ol>
</statement>
<hint>
<p>
(a) Not a group; (c) a group.
</p>
</hint>
</exercise>
<exercise number="3" xml:id="exercise-groups-rectangle-symmetries">
<statement>
<p>
Write out Cayley tables for groups formed by the symmetries of a rectangle and for <m>({\mathbb Z}_4, +)</m>.
How many elements are in each group?
Are the groups the same?
Why or why not?
</p>
</statement>
</exercise>
<exercise number="4">
<statement>
<p>
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?
</p>
</statement>
</exercise>
<exercise number="5">
<statement>
<p>
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 <m>D_4</m>.
</p>
</statement>
</exercise>
<exercise number="6">
<statement>
<p>
Give a multiplication table for the group <m>U(12)</m>.
</p>
</statement>
<hint>
<p>
<me>
\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}
</me>
</p>
</hint>
</exercise>
<exercise number="7">
<statement>
<p>
Let <m>S = {\mathbb R} \setminus \{ -1 \}</m> and define a binary operation on <m>S</m> by <m>a \ast b = a + b + ab</m>.
Prove that <m>(S, \ast)</m> is an abelian group.
</p>
</statement>
</exercise>
<exercise number="8">
<statement>
<p>
Give an example of two elements <m>A</m> and <m>B</m> in
<m>GL_2({\mathbb R})</m> with <m>AB \neq BA</m>.
</p>
</statement>
<hint>
<p>
Pick two matrices.
Almost any pair will work.
</p>
</hint>
</exercise>
<exercise number="9">
<statement>
<p>
Prove that the product of two matrices in <m>SL_2({\mathbb R})</m> has determinant one.
</p>
</statement>
</exercise>
<exercise number="10">
<statement>
<p>
Prove that the set of matrices of the form
<me>
\begin{pmatrix} 1 & x & y \\ 0 & 1 & z \\ 0 & 0 & 1 \end{pmatrix}
</me>
is a group under matrix multiplication.
This group, known as the <term>Heisenberg group</term>,
is important in quantum physics.
Matrix multiplication in the Heisenberg group is defined by
<me>
\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}
</me>.
</p>
</statement>
</exercise>
<exercise number="11">
<statement>
<p>
Prove that <m>\det(AB) = \det(A) \det(B)</m> in <m>GL_2({\mathbb R})</m>.
Use this result to show that the binary operation in the group <m>GL_2({\mathbb R})</m> is closed;
that is, if <m>A</m> and <m>B</m> are in <m>GL_2({\mathbb R})</m>,
then <m>AB \in GL_2({\mathbb R})</m>.
</p>
</statement>
</exercise>
<exercise number="12">
<statement>
<p>
Let <m>{\mathbb Z}_2^n = \{ (a_1, a_2, \ldots,
a_n) : a_i \in {\mathbb Z}_2 \}</m>.
Define a binary operation on <m>{\mathbb Z}_2^n</m> by
<me>
(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)
</me>.
Prove that <m>{\mathbb Z}_2^n</m> is a group under this operation.
This group is important in algebraic coding theory.
</p>
</statement>
</exercise>
<exercise number="13">
<statement>
<p>
Show that <m>{\mathbb R}^{\ast} = {\mathbb R} \setminus \{0 \}</m> is a group under the operation of multiplication.
</p>
</statement>
</exercise>
<exercise number="14">
<statement>
<p>
Given the groups <m>{\mathbb R}^{\ast}</m> and <m>{\mathbb Z}</m>,
let <m>G = {\mathbb R}^{\ast} \times {\mathbb Z}</m>.
Define a binary operation <m>\circ</m> on <m>G</m> by <m>(a,m) \circ (b,n) = (ab,
m + n)</m>.
Show that <m>G</m> is a group under this operation.
</p>
</statement>
</exercise>
<exercise number="15">
<statement>
<p>
Prove or disprove that every group containing six elements is abelian.
</p>
</statement>
<hint>
<p>
There is a nonabelian group containing six elements.
</p>
</hint>
</exercise>
<exercise number="16">
<statement>
<p>
Give a specific example of some group <m>G</m> and elements
<m>g,
h \in G</m> where <m>(gh)^n \neq g^nh^n</m>.
</p>
</statement>
<hint>
<p>
Look at the symmetry group of an equilateral triangle or a square.
</p>
</hint>
</exercise>
<exercise number="17">
<statement>
<p>
Give an example of three different groups with eight elements.
Why are the groups different?
</p>
</statement>
<hint>
<p>
The are five different groups of order 8.
</p>
</hint>
</exercise>
<exercise number="18">
<statement>
<p>
Show that there are <m>n!</m> permutations of a set containing <m>n</m> items.
</p>
</statement>
<hint>
<p>
Let
<me>
\sigma = \begin{pmatrix} 1 & 2 & \cdots & n \\ a_1 & a_2 & \cdots & a_n \end{pmatrix}
</me>
be in <m>S_n</m>.
All of the <m>a_i</m>s must be distinct.
There are <m>n</m> ways to choose <m>a_1</m>,
<m>n-1</m> ways to choose <m>a_2</m>,
<m>\ldots</m>, 2 ways to choose <m>a_{n - 1}</m>,
and only one way to choose <m>a_n</m>.
Therefore, we can form <m>\sigma</m> in <m>n(n - 1) \cdots 2 \cdot 1 = n!</m> ways.
</p>
</hint>
</exercise>
<exercise number="19">
<statement>
<p>
Show that
<me>
0 + a \equiv a + 0 \equiv a \pmod{ n }
</me>
for all <m>a \in {\mathbb Z}_n</m>.
</p>
</statement>
</exercise>
<exercise number="20">
<statement>
<p>
Prove that there is a multiplicative identity for the integers modulo <m>n</m>:
<me>
a \cdot 1 \equiv a \pmod{n}
</me>.
</p>
</statement>
</exercise>
<exercise number="21">
<statement>
<p>
For each <m>a \in {\mathbb Z}_n</m> find an element <m>b \in {\mathbb Z}_n</m> such that
<me>
a + b \equiv b + a \equiv 0 \pmod{ n}
</me>.
</p>
</statement>
</exercise>
<exercise number="22">
<statement>
<p>
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 <m>n</m>.
</p>
</statement>
</exercise>
<exercise number="23">
<statement>
<p>
Show that addition and multiplication mod <m>n</m> are associative operations.
</p>
</statement>
</exercise>
<exercise number="24">
<statement>
<p>
Show that multiplication distributes over addition modulo <m>n</m>:
<me>
a(b + c) \equiv ab + ac \pmod{n}
</me>.
</p>
</statement>
</exercise>
<exercise number="25">
<statement>
<p>
Let <m>a</m> and <m>b</m> be elements in a group <m>G</m>.
Prove that <m>ab^na^{-1} = (aba^{-1})^n</m> for <m>n \in \mathbb Z</m>.
</p>
</statement>
<hint>
<p>
<md>
<mrow>(aba^{-1})^n & = (aba^{-1})(aba^{-1}) \cdots (aba^{-1})</mrow>
<mrow>& = ab(aa^{-1})b(aa^{-1})b \cdots b(aa^{-1})ba^{-1}</mrow>
<mrow>& = ab^na^{-1}</mrow>
</md>.
</p>
</hint>
</exercise>
<exercise number="26">
<statement>
<p>
Let <m>U(n)</m> be the group of units in <m>{\mathbb Z}_n</m>.
If <m>n \gt 2</m>, prove that there is an element
<m>k \in U(n)</m> such that <m>k^2 = 1</m> and <m>k \neq 1</m>.
</p>
</statement>
</exercise>
<exercise number="27">
<statement>
<p>
Prove that the inverse of <m>g _1 g_2 \cdots g_n</m> is <m>g_n^{-1} g_{n-1}^{-1} \cdots g_1^{-1}</m>.
</p>
</statement>
</exercise>
<exercise number="28">
<statement>
<p>
Prove the remainder of <xref ref="proposition-group-equations" />:
if <m>G</m> is a group and <m>a, b \in G</m>,
then the equation <m>xa = b</m> has a unique solution in <m>G</m>.
</p>
</statement>
</exercise>
<exercise number="29">
<statement>
<p>
Prove <xref ref="theorem-exponent-laws" />.
</p>
</statement>
</exercise>
<exercise number="30">
<statement>
<p>
Prove the right and left cancellation laws for a group <m>G</m>;
that is, show that in the group <m>G</m>,
<m>ba = ca</m> implies <m>b = c</m> and <m>ab = ac</m> implies <m>b = c</m> for elements <m>a,
b, c \in G</m>.
</p>
</statement>
</exercise>
<exercise number="31">
<statement>
<p>
Show that if <m>a^2 = e</m> for all elements <m>a</m> in a group <m>G</m>,
then <m>G</m> must be abelian.
</p>
</statement>
<hint>
<p>
Since <m>abab = (ab)^2 = e = a^2 b^2 = aabb</m>,
we know that <m>ba = ab</m>.
</p>
</hint>
</exercise>
<exercise number="32">
<statement>
<p>
Show that if <m>G</m> is a finite group of even order,
then there is an <m>a \in G</m> such that <m>a</m> is not the identity and <m>a^2 = e</m>.
</p>
</statement>
</exercise>
<exercise number="33">
<statement>
<p>
Let <m>G</m> be a group and suppose that
<m>(ab)^2 = a^2b^2</m> for all <m>a</m> and <m>b</m> in <m>G</m>.
Prove that <m>G</m> is an abelian group.
</p>
</statement>
</exercise>
<exercise number="34">
<statement>
<p>
Find all the subgroups of <m>{\mathbb Z}_3 \times {\mathbb Z}_3</m>.
Use this information to show that
<m>{\mathbb Z}_3 \times {\mathbb Z}_3</m> is not the same group as <m>{\mathbb Z}_9</m>.
(See <xref ref="example-groups-z2xz2" /> for a short description of the product of groups.)
</p>
</statement>
</exercise>
<exercise number="35">
<statement>
<p>
Find all the subgroups of the symmetry group of an equilateral triangle.
</p>
</statement>
<hint>
<p>
<m>H_1 = \{ id \}</m>, <m>H_2 = \{ id, \rho_1, \rho_2 \}</m>,
<m>H_3 = \{ id, \mu_1 \}</m>,
<m>H_4 = \{ id, \mu_2 \}</m>,
<m>H_5 = \{ id, \mu_3 \}</m>, <m>S_3</m>.
</p>
</hint>
</exercise>
<exercise number="36">
<statement>
<p>
Compute the subgroups of the symmetry group of a square.
</p>
</statement>
</exercise>
<exercise number="37">
<statement>
<p>
Let <m>H = \{2^k : k \in {\mathbb Z} \}</m>.
Show that <m>H</m> is a subgroup of <m>{\mathbb Q}^*</m>.
</p>
</statement>
</exercise>
<exercise number="38">
<statement>
<p>
Let <m>n = 0, 1, 2, \ldots</m> and <m>n {\mathbb Z} = \{ nk : k \in {\mathbb Z} \}</m>.
Prove that <m>n {\mathbb Z}</m> is a subgroup of <m>{\mathbb Z}</m>.
Show that these subgroups are the only subgroups of <m>\mathbb{Z}</m>.
</p>
</statement>
</exercise>
<exercise number="39" xml:id="exercise-groups-circle-group">
<statement>
<p>
Let <m>{\mathbb T} = \{ z \in {\mathbb C}^* : |z| =1 \}</m>.
Prove that <m>{\mathbb T}</m> is a subgroup of <m>{\mathbb C}^*</m>.
</p>
</statement>
</exercise>
<exercise number="40">
<statement>
<p>
<me>
\begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}
</me>
where <m>\theta \in {\mathbb R}</m>.
Prove that <m>G</m> is a subgroup of <m>SL_2({\mathbb R})</m>.
</p>
</statement>
</exercise>
<exercise number="41">
<statement>
<p>
Prove that
<me>
G = \{ a + b \sqrt{2} : a, b \in {\mathbb Q} \text{ and } a \text{ and } b \text{ are not both zero} \}
</me>
is a subgroup of <m>{\mathbb R}^{\ast}</m> under the group operation of multiplication.
</p>
</statement>
<hint>
<p>
The identity of <m>G</m> is <m>1 = 1 + 0 \sqrt{2}</m>.
Since <m>(a + b \sqrt{2}\, )(c + d \sqrt{2}\, ) = (ac + 2bd) + (ad + bc)\sqrt{2}</m>,
<m>G</m> is closed under multiplication.
Finally, <m>(a + b \sqrt{2}\, )^{-1} = a/(a^2 - 2b^2) - b\sqrt{2}/(a^2 - 2 b^2)</m>.
</p>
</hint>
</exercise>
<exercise number="42">
<statement>
<p>
Let <m>G</m> be the group of
<m>2 \times 2</m> matrices under addition and
<me>
H = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} : a + d = 0 \right\}
</me>.
Prove that <m>H</m> is a subgroup of <m>G</m>.
</p>
</statement>
</exercise>
<exercise number="43">
<statement>
<p>
Prove or disprove: <m>SL_2( {\mathbb Z} )</m>,
the set of <m>2 \times 2</m> matrices with integer entries and determinant one,
is a subgroup of <m>SL_2( {\mathbb R} )</m>.
</p>
</statement>
</exercise>
<exercise number="44">
<statement>
<p>
List the subgroups of the quaternion group, <m>Q_8</m>.
</p>
</statement>
</exercise>
<exercise number="45">
<statement>
<p>
Prove that the intersection of two subgroups of a group <m>G</m> is also a subgroup of <m>G</m>.
</p>
</statement>
</exercise>
<exercise number="46">
<statement>
<p>
Prove or disprove: If <m>H</m> and <m>K</m> are subgroups of a group <m>G</m>,
then <m>H \cup K</m> is a subgroup of <m>G</m>.
</p>
</statement>
<hint>
<p>
Look at <m>S_3</m>.
</p>
</hint>
</exercise>
<exercise number="47">
<statement>
<p>
Prove or disprove: If <m>H</m> and <m>K</m> are subgroups of a group <m>G</m>,
then <m>H K = \{hk : h \in H \text{ and } k \in K \}</m> is a subgroup of <m>G</m>.
What if <m>G</m> is abelian?
</p>
</statement>
</exercise>
<exercise number="48">
<statement>
<p>
Let <m>G</m> be a group and <m>g \in G</m>.
Show that
<me>
Z(G) = \{ x \in G : gx = xg \text{ for all } g \in G \}
</me>
is a subgroup of <m>G</m>.
This subgroup is called the <term>center</term> of <m>G</m>.
<notation>
<usage><m>Z(G)</m></usage>
<description>the center of a group</description>
</notation>
</p>
</statement>
</exercise>
<exercise number="49">
<statement>
<p>
Let <m>a</m> and <m>b</m> be elements of a group <m>G</m>.
If <m>a^4b = ba</m> and <m>a^3 = e</m>,
prove that <m>ab = ba</m>.
</p>
</statement>
<hint>
<p>
Since <m>a^4b = ba</m>,
it must be the case that <m>b = a^6 b = a^2 b a</m>,
and we can conclude that <m> ab = a^3 b a = ba</m>.
</p>
</hint>
</exercise>
<exercise number="50">
<statement>
<p>
Give an example of an infinite group in which every nontrivial subgroup is infinite.
</p>
</statement>
</exercise>
<exercise number="51">
<statement>
<p>
If <m>xy = x^{-1} y^{-1}</m> for all <m>x</m> and <m>y</m> in <m>G</m>,
prove that <m>G</m> must be abelian.
</p>
</statement>
</exercise>
<exercise number="52">
<statement>
<p>
Prove or disprove: Every proper subgroup of an nonabelian group is nonabelian.
</p>
</statement>
</exercise>
<exercise number="53">
<statement>
<p>
Let <m>H</m> be a subgroup of <m>G</m> and
<me>
C(H) = \{ g \in G : gh = hg \text{ for all } h \in H \}
</me>.
Prove <m>C(H)</m> is a subgroup of <m>G</m>.
This subgroup is called the <term>centralizer</term> of <m>H</m> in <m>G</m>.
</p>
</statement>
</exercise>
<exercise number="54">
<statement>
<p>
Let <m>H</m> be a subgroup of <m>G</m>.
If <m>g \in G</m>, show that
<m>gHg^{-1} = \{g^{-1}hg : h\in H\}</m> is also a subgroup of <m>G</m>.
</p>
</statement>
</exercise>
<exercisegroup cols="2">
<introduction>
<p>
In each group, how many solutions are there to <m>x^2=e</m>?
</p>
</introduction>
<exercise number="55">
<statement>
<p>
<m>C_n</m>, <m>n</m> odd.
</p>
</statement>
<answer>
<p>
<m>1</m>
</p>
</answer>
</exercise>
<exercise number="56">
<statement>
<p>
<m>C_n</m>, <m>n</m> even.
</p>
</statement>
<answer>
<p>
<m>2</m>
</p>
</answer>
</exercise>
<exercise number="57">
<statement>
<p>
<m>D_n</m>, <m>n</m> odd.
</p>
</statement>
<answer>
<p>
<m>n</m>
</p>
</answer>
</exercise>
<exercise number="58">
<statement>
<p>
<m>D_n</m>, <m>n</m> even.
</p>
</statement>
<answer>
<p>
<m>n+1</m>
</p>
</answer>
</exercise>
</exercisegroup>
<exercise number="59">
<introduction>
<p>
This is an odd-numbered exercise with tasks.
</p>
</introduction>
<task>
<statement>
<p>
What is <m>1+1</m>?
</p>
</statement>
<answer>
<p>
<m>2</m>
</p>
</answer>
</task>
<task>
<introduction>
<p>
This task has subtasks.
</p>
</introduction>
<task>
<statement>
<p>
What is <m>3+3</m>?
</p>
</statement>
<answer>
<p>
<m>6</m>
</p>
</answer>
</task>
<task>
<statement>
<p>
What is <m>5+5</m>?
</p>
</statement>
<answer>
<p>
<m>10</m>
</p>
</answer>
</task>
</task>
</exercise>
<exercise number="60">
<introduction>
<p>
This is an even-numbered exercise with tasks.
</p>
</introduction>
<task>
<statement>
<p>
What is <m>2+2</m>?
</p>
</statement>
<answer>
<p>
<m>4</m>
</p>
</answer>
</task>
<task>
<introduction>
<p>
This task has subtasks.
</p>
</introduction>
<task>
<statement>
<p>
What is <m>4+4</m>?
</p>
</statement>
<answer>
<p>
<m>8</m>
</p>
</answer>
</task>
<task>
<statement>
<p>
What is <m>6+6</m>?
</p>
</statement>
<answer>
<p>
<m>12</m>
</p>
</answer>
</task>
</task>
</exercise>
</exercises>
Exercises 1.5 Exercises
View Source for exercises
1.
View Source for exercise
<exercise number="1">
<statement>
<p>
Find all <m>x \in {\mathbb Z}</m> satisfying each of the following equations.
</p>
<ol cols="2">
<li>
<p>
<m>3x \equiv 2 \pmod{7}</m>
</p>
</li>
<li>
<p>
<m>5x + 1 \equiv 13 \pmod{23}</m>
</p>
</li>
<li>
<p>
<m>5x + 1 \equiv 13 \pmod{26}</m>
</p>
</li>
<li>
<p>
<m>9x \equiv 3 \pmod{5}</m>
</p>
</li>
<li>
<p>
<m>5x \equiv 1 \pmod{6}</m>
</p>
</li>
<li>
<p>
<m>3x \equiv 1 \pmod{6}</m>
</p>
</li>
</ol>
</statement>
<hint>
<p>
(a) <m>3 + 7 \mathbb Z = \{ \ldots, -4, 3, 10, \ldots \}</m>; (c)
<m>18 + 26 \mathbb Z</m>; (e) <m>5 + 6 \mathbb Z</m>.
</p>
</hint>
</exercise>
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.
View Source for hint
<hint>
<p>
(a) <m>3 + 7 \mathbb Z = \{ \ldots, -4, 3, 10, \ldots \}</m>; (c)
<m>18 + 26 \mathbb Z</m>; (e) <m>5 + 6 \mathbb Z</m>.
</p>
</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.
View Source for exercise
<exercise number="2">
<statement>
<p>
Which of the following multiplication tables defined on the set <m>G = \{ a, b, c, d \}</m> form a group?
Support your answer in each case.
</p>
<ol cols="2">
<li>
<p>
<me>
\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}
</me>
</p>
</li>
<li>
<p>
<me>
\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}
</me>
</p>
</li>
<li>
<p>
<me>
\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}
</me>
</p>
</li>
<li>
<p>
<me>
\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}
</me>
</p>
</li>
</ol>
</statement>
<hint>
<p>
(a) Not a group; (c) a group.
</p>
</hint>
</exercise>
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.
View Source for hint
<hint>
<p>
(a) Not a group; (c) a group.
</p>
</hint>
(a) Not a group; (c) a group.
3.
View Source for exercise
<exercise number="3" xml:id="exercise-groups-rectangle-symmetries">
<statement>
<p>
Write out Cayley tables for groups formed by the symmetries of a rectangle and for <m>({\mathbb Z}_4, +)</m>.
How many elements are in each group?
Are the groups the same?
Why or why not?
</p>
</statement>
</exercise>
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.
View Source for exercise
<exercise number="4">
<statement>
<p>
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?
</p>
</statement>
</exercise>
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.
View Source for exercise
<exercise number="5">
<statement>
<p>
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 <m>D_4</m>.
</p>
</statement>
</exercise>
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.
View Source for exercise
<exercise number="6">
<statement>
<p>
Give a multiplication table for the group <m>U(12)</m>.
</p>
</statement>
<hint>
<p>
<me>
\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}
</me>
</p>
</hint>
</exercise>
Give a multiplication table for the group \(U(12)\text{.}\)
Hint.
View Source for hint
<hint>
<p>
<me>
\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}
</me>
</p>
</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.
View Source for exercise
<exercise number="7">
<statement>
<p>
Let <m>S = {\mathbb R} \setminus \{ -1 \}</m> and define a binary operation on <m>S</m> by <m>a \ast b = a + b + ab</m>.
Prove that <m>(S, \ast)</m> is an abelian group.
</p>
</statement>
</exercise>
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.
View Source for exercise
<exercise number="8">
<statement>
<p>
Give an example of two elements <m>A</m> and <m>B</m> in
<m>GL_2({\mathbb R})</m> with <m>AB \neq BA</m>.
</p>
</statement>
<hint>
<p>
Pick two matrices.
Almost any pair will work.
</p>
</hint>
</exercise>
Give an example of two elements \(A\) and \(B\) in \(GL_2({\mathbb R})\) with \(AB \neq BA\text{.}\)
Hint.
View Source for hint
<hint>
<p>
Pick two matrices.
Almost any pair will work.
</p>
</hint>
Pick two matrices. Almost any pair will work.
9.
View Source for exercise
<exercise number="9">
<statement>
<p>
Prove that the product of two matrices in <m>SL_2({\mathbb R})</m> has determinant one.
</p>
</statement>
</exercise>
Prove that the product of two matrices in \(SL_2({\mathbb R})\) has determinant one.
10.
View Source for exercise
<exercise number="10">
<statement>
<p>
Prove that the set of matrices of the form
<me>
\begin{pmatrix} 1 & x & y \\ 0 & 1 & z \\ 0 & 0 & 1 \end{pmatrix}
</me>
is a group under matrix multiplication.
This group, known as the <term>Heisenberg group</term>,
is important in quantum physics.
Matrix multiplication in the Heisenberg group is defined by
<me>
\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}
</me>.
</p>
</statement>
</exercise>
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}\text{.}
\end{equation*}
11.
View Source for exercise
<exercise number="11">
<statement>
<p>
Prove that <m>\det(AB) = \det(A) \det(B)</m> in <m>GL_2({\mathbb R})</m>.
Use this result to show that the binary operation in the group <m>GL_2({\mathbb R})</m> is closed;
that is, if <m>A</m> and <m>B</m> are in <m>GL_2({\mathbb R})</m>,
then <m>AB \in GL_2({\mathbb R})</m>.
</p>
</statement>
</exercise>
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.
View Source for exercise
<exercise number="12">
<statement>
<p>
Let <m>{\mathbb Z}_2^n = \{ (a_1, a_2, \ldots,
a_n) : a_i \in {\mathbb Z}_2 \}</m>.
Define a binary operation on <m>{\mathbb Z}_2^n</m> by
<me>
(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)
</me>.
Prove that <m>{\mathbb Z}_2^n</m> is a group under this operation.
This group is important in algebraic coding theory.
</p>
</statement>
</exercise>
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)\text{.}
\end{equation*}
Prove that \({\mathbb Z}_2^n\) is a group under this operation. This group is important in algebraic coding theory.
13.
View Source for exercise
<exercise number="13">
<statement>
<p>
Show that <m>{\mathbb R}^{\ast} = {\mathbb R} \setminus \{0 \}</m> is a group under the operation of multiplication.
</p>
</statement>
</exercise>
Show that \({\mathbb R}^{\ast} = {\mathbb R} \setminus \{0 \}\) is a group under the operation of multiplication.
14.
View Source for exercise
<exercise number="14">
<statement>
<p>
Given the groups <m>{\mathbb R}^{\ast}</m> and <m>{\mathbb Z}</m>,
let <m>G = {\mathbb R}^{\ast} \times {\mathbb Z}</m>.
Define a binary operation <m>\circ</m> on <m>G</m> by <m>(a,m) \circ (b,n) = (ab,
m + n)</m>.
Show that <m>G</m> is a group under this operation.
</p>
</statement>
</exercise>
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.
View Source for exercise
<exercise number="15">
<statement>
<p>
Prove or disprove that every group containing six elements is abelian.
</p>
</statement>
<hint>
<p>
There is a nonabelian group containing six elements.
</p>
</hint>
</exercise>
Prove or disprove that every group containing six elements is abelian.
Hint.
View Source for hint
<hint>
<p>
There is a nonabelian group containing six elements.
</p>
</hint>
There is a nonabelian group containing six elements.
16.
View Source for exercise
<exercise number="16">
<statement>
<p>
Give a specific example of some group <m>G</m> and elements
<m>g,
h \in G</m> where <m>(gh)^n \neq g^nh^n</m>.
</p>
</statement>
<hint>
<p>
Look at the symmetry group of an equilateral triangle or a square.
</p>
</hint>
</exercise>
Give a specific example of some group \(G\) and elements \(g,
h \in G\) where \((gh)^n \neq g^nh^n\text{.}\)
Hint.
View Source for hint
<hint>
<p>
Look at the symmetry group of an equilateral triangle or a square.
</p>
</hint>
Look at the symmetry group of an equilateral triangle or a square.
17.
View Source for exercise
<exercise number="17">
<statement>
<p>
Give an example of three different groups with eight elements.
Why are the groups different?
</p>
</statement>
<hint>
<p>
The are five different groups of order 8.
</p>
</hint>
</exercise>
Give an example of three different groups with eight elements. Why are the groups different?
Hint.
View Source for hint
<hint>
<p>
The are five different groups of order 8.
</p>
</hint>
The are five different groups of order 8.
18.
View Source for exercise
<exercise number="18">
<statement>
<p>
Show that there are <m>n!</m> permutations of a set containing <m>n</m> items.
</p>
</statement>
<hint>
<p>
Let
<me>
\sigma = \begin{pmatrix} 1 & 2 & \cdots & n \\ a_1 & a_2 & \cdots & a_n \end{pmatrix}
</me>
be in <m>S_n</m>.
All of the <m>a_i</m>s must be distinct.
There are <m>n</m> ways to choose <m>a_1</m>,
<m>n-1</m> ways to choose <m>a_2</m>,
<m>\ldots</m>, 2 ways to choose <m>a_{n - 1}</m>,
and only one way to choose <m>a_n</m>.
Therefore, we can form <m>\sigma</m> in <m>n(n - 1) \cdots 2 \cdot 1 = n!</m> ways.
</p>
</hint>
</exercise>
Show that there are \(n!\) permutations of a set containing \(n\) items.
Hint.
View Source for hint
<hint>
<p>
Let
<me>
\sigma = \begin{pmatrix} 1 & 2 & \cdots & n \\ a_1 & a_2 & \cdots & a_n \end{pmatrix}
</me>
be in <m>S_n</m>.
All of the <m>a_i</m>s must be distinct.
There are <m>n</m> ways to choose <m>a_1</m>,
<m>n-1</m> ways to choose <m>a_2</m>,
<m>\ldots</m>, 2 ways to choose <m>a_{n - 1}</m>,
and only one way to choose <m>a_n</m>.
Therefore, we can form <m>\sigma</m> in <m>n(n - 1) \cdots 2 \cdot 1 = n!</m> ways.
</p>
</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.
View Source for exercise
<exercise number="19">
<statement>
<p>
Show that
<me>
0 + a \equiv a + 0 \equiv a \pmod{ n }
</me>
for all <m>a \in {\mathbb Z}_n</m>.
</p>
</statement>
</exercise>
Show that
\begin{equation*}
0 + a \equiv a + 0 \equiv a \pmod{ n }
\end{equation*}
for all \(a \in {\mathbb Z}_n\text{.}\)
20.
View Source for exercise
<exercise number="20">
<statement>
<p>
Prove that there is a multiplicative identity for the integers modulo <m>n</m>:
<me>
a \cdot 1 \equiv a \pmod{n}
</me>.
</p>
</statement>
</exercise>
Prove that there is a multiplicative identity for the integers modulo \(n\text{:}\)
\begin{equation*}
a \cdot 1 \equiv a \pmod{n}\text{.}
\end{equation*}
21.
View Source for exercise
<exercise number="21">
<statement>
<p>
For each <m>a \in {\mathbb Z}_n</m> find an element <m>b \in {\mathbb Z}_n</m> such that
<me>
a + b \equiv b + a \equiv 0 \pmod{ n}
</me>.
</p>
</statement>
</exercise>
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}\text{.}
\end{equation*}
22.
View Source for exercise
<exercise number="22">
<statement>
<p>
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 <m>n</m>.
</p>
</statement>
</exercise>
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.
View Source for exercise
<exercise number="23">
<statement>
<p>
Show that addition and multiplication mod <m>n</m> are associative operations.
</p>
</statement>
</exercise>
Show that addition and multiplication mod \(n\) are associative operations.
24.
View Source for exercise
<exercise number="24">
<statement>
<p>
Show that multiplication distributes over addition modulo <m>n</m>:
<me>
a(b + c) \equiv ab + ac \pmod{n}
</me>.
</p>
</statement>
</exercise>
Show that multiplication distributes over addition modulo \(n\text{:}\)
\begin{equation*}
a(b + c) \equiv ab + ac \pmod{n}\text{.}
\end{equation*}
25.
View Source for exercise
<exercise number="25">
<statement>
<p>
Let <m>a</m> and <m>b</m> be elements in a group <m>G</m>.
Prove that <m>ab^na^{-1} = (aba^{-1})^n</m> for <m>n \in \mathbb Z</m>.
</p>
</statement>
<hint>
<p>
<md>
<mrow>(aba^{-1})^n & = (aba^{-1})(aba^{-1}) \cdots (aba^{-1})</mrow>
<mrow>& = ab(aa^{-1})b(aa^{-1})b \cdots b(aa^{-1})ba^{-1}</mrow>
<mrow>& = ab^na^{-1}</mrow>
</md>.
</p>
</hint>
</exercise>
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.
View Source for hint
<hint>
<p>
<md>
<mrow>(aba^{-1})^n & = (aba^{-1})(aba^{-1}) \cdots (aba^{-1})</mrow>
<mrow>& = ab(aa^{-1})b(aa^{-1})b \cdots b(aa^{-1})ba^{-1}</mrow>
<mrow>& = ab^na^{-1}</mrow>
</md>.
</p>
</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}\text{.}
\end{align*}
26.
View Source for exercise
<exercise number="26">
<statement>
<p>
Let <m>U(n)</m> be the group of units in <m>{\mathbb Z}_n</m>.
If <m>n \gt 2</m>, prove that there is an element
<m>k \in U(n)</m> such that <m>k^2 = 1</m> and <m>k \neq 1</m>.
</p>
</statement>
</exercise>
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.
View Source for exercise
<exercise number="27">
<statement>
<p>
Prove that the inverse of <m>g _1 g_2 \cdots g_n</m> is <m>g_n^{-1} g_{n-1}^{-1} \cdots g_1^{-1}</m>.
</p>
</statement>
</exercise>
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.
View Source for exercise
<exercise number="28">
<statement>
<p>
Prove the remainder of <xref ref="proposition-group-equations" />:
if <m>G</m> is a group and <m>a, b \in G</m>,
then the equation <m>xa = b</m> has a unique solution in <m>G</m>.
</p>
</statement>
</exercise>
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.
View Source for exercise
<exercise number="29">
<statement>
<p>
Prove <xref ref="theorem-exponent-laws" />.
</p>
</statement>
</exercise>
Prove Theorem 1.2.16.
30.
View Source for exercise
<exercise number="30">
<statement>
<p>
Prove the right and left cancellation laws for a group <m>G</m>;
that is, show that in the group <m>G</m>,
<m>ba = ca</m> implies <m>b = c</m> and <m>ab = ac</m> implies <m>b = c</m> for elements <m>a,
b, c \in G</m>.
</p>
</statement>
</exercise>
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.
View Source for exercise
<exercise number="31">
<statement>
<p>
Show that if <m>a^2 = e</m> for all elements <m>a</m> in a group <m>G</m>,
then <m>G</m> must be abelian.
</p>
</statement>
<hint>
<p>
Since <m>abab = (ab)^2 = e = a^2 b^2 = aabb</m>,
we know that <m>ba = ab</m>.
</p>
</hint>
</exercise>
Show that if \(a^2 = e\) for all elements \(a\) in a group \(G\text{,}\) then \(G\) must be abelian.
Hint.
View Source for hint
<hint>
<p>
Since <m>abab = (ab)^2 = e = a^2 b^2 = aabb</m>,
we know that <m>ba = ab</m>.
</p>
</hint>
Since \(abab = (ab)^2 = e = a^2 b^2 = aabb\text{,}\) we know that \(ba = ab\text{.}\)
32.
View Source for exercise
<exercise number="32">
<statement>
<p>
Show that if <m>G</m> is a finite group of even order,
then there is an <m>a \in G</m> such that <m>a</m> is not the identity and <m>a^2 = e</m>.
</p>
</statement>
</exercise>
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.
View Source for exercise
<exercise number="33">
<statement>
<p>
Let <m>G</m> be a group and suppose that
<m>(ab)^2 = a^2b^2</m> for all <m>a</m> and <m>b</m> in <m>G</m>.
Prove that <m>G</m> is an abelian group.
</p>
</statement>
</exercise>
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.
View Source for exercise
<exercise number="34">
<statement>
<p>
Find all the subgroups of <m>{\mathbb Z}_3 \times {\mathbb Z}_3</m>.
Use this information to show that
<m>{\mathbb Z}_3 \times {\mathbb Z}_3</m> is not the same group as <m>{\mathbb Z}_9</m>.
(See <xref ref="example-groups-z2xz2" /> for a short description of the product of groups.)
</p>
</statement>
</exercise>
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.
View Source for exercise
<exercise number="35">
<statement>
<p>
Find all the subgroups of the symmetry group of an equilateral triangle.
</p>
</statement>
<hint>
<p>
<m>H_1 = \{ id \}</m>, <m>H_2 = \{ id, \rho_1, \rho_2 \}</m>,
<m>H_3 = \{ id, \mu_1 \}</m>,
<m>H_4 = \{ id, \mu_2 \}</m>,
<m>H_5 = \{ id, \mu_3 \}</m>, <m>S_3</m>.
</p>
</hint>
</exercise>
Find all the subgroups of the symmetry group of an equilateral triangle.
Hint.
View Source for hint
<hint>
<p>
<m>H_1 = \{ id \}</m>, <m>H_2 = \{ id, \rho_1, \rho_2 \}</m>,
<m>H_3 = \{ id, \mu_1 \}</m>,
<m>H_4 = \{ id, \mu_2 \}</m>,
<m>H_5 = \{ id, \mu_3 \}</m>, <m>S_3</m>.
</p>
</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.
View Source for exercise
<exercise number="36">
<statement>
<p>
Compute the subgroups of the symmetry group of a square.
</p>
</statement>
</exercise>
Compute the subgroups of the symmetry group of a square.
37.
View Source for exercise
<exercise number="37">
<statement>
<p>
Let <m>H = \{2^k : k \in {\mathbb Z} \}</m>.
Show that <m>H</m> is a subgroup of <m>{\mathbb Q}^*</m>.
</p>
</statement>
</exercise>
Let \(H = \{2^k : k \in {\mathbb Z} \}\text{.}\) Show that \(H\) is a subgroup of \({\mathbb Q}^*\text{.}\)
38.
View Source for exercise
<exercise number="38">
<statement>
<p>
Let <m>n = 0, 1, 2, \ldots</m> and <m>n {\mathbb Z} = \{ nk : k \in {\mathbb Z} \}</m>.
Prove that <m>n {\mathbb Z}</m> is a subgroup of <m>{\mathbb Z}</m>.
Show that these subgroups are the only subgroups of <m>\mathbb{Z}</m>.
</p>
</statement>
</exercise>
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.
View Source for exercise
<exercise number="39" xml:id="exercise-groups-circle-group">
<statement>
<p>
Let <m>{\mathbb T} = \{ z \in {\mathbb C}^* : |z| =1 \}</m>.
Prove that <m>{\mathbb T}</m> is a subgroup of <m>{\mathbb C}^*</m>.
</p>
</statement>
</exercise>
Let \({\mathbb T} = \{ z \in {\mathbb C}^* : |z| =1 \}\text{.}\) Prove that \({\mathbb T}\) is a subgroup of \({\mathbb C}^*\text{.}\)
40.
View Source for exercise
<exercise number="40">
<statement>
<p>
<me>
\begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}
</me>
where <m>\theta \in {\mathbb R}</m>.
Prove that <m>G</m> is a subgroup of <m>SL_2({\mathbb R})</m>.
</p>
</statement>
</exercise>
\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.
View Source for exercise
<exercise number="41">
<statement>
<p>
Prove that
<me>
G = \{ a + b \sqrt{2} : a, b \in {\mathbb Q} \text{ and } a \text{ and } b \text{ are not both zero} \}
</me>
is a subgroup of <m>{\mathbb R}^{\ast}</m> under the group operation of multiplication.
</p>
</statement>
<hint>
<p>
The identity of <m>G</m> is <m>1 = 1 + 0 \sqrt{2}</m>.
Since <m>(a + b \sqrt{2}\, )(c + d \sqrt{2}\, ) = (ac + 2bd) + (ad + bc)\sqrt{2}</m>,
<m>G</m> is closed under multiplication.
Finally, <m>(a + b \sqrt{2}\, )^{-1} = a/(a^2 - 2b^2) - b\sqrt{2}/(a^2 - 2 b^2)</m>.
</p>
</hint>
</exercise>
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.
View Source for hint
<hint>
<p>
The identity of <m>G</m> is <m>1 = 1 + 0 \sqrt{2}</m>.
Since <m>(a + b \sqrt{2}\, )(c + d \sqrt{2}\, ) = (ac + 2bd) + (ad + bc)\sqrt{2}</m>,
<m>G</m> is closed under multiplication.
Finally, <m>(a + b \sqrt{2}\, )^{-1} = a/(a^2 - 2b^2) - b\sqrt{2}/(a^2 - 2 b^2)</m>.
</p>
</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.
View Source for exercise
<exercise number="42">
<statement>
<p>
Let <m>G</m> be the group of
<m>2 \times 2</m> matrices under addition and
<me>
H = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} : a + d = 0 \right\}
</me>.
Prove that <m>H</m> is a subgroup of <m>G</m>.
</p>
</statement>
</exercise>
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\}\text{.}
\end{equation*}
Prove that \(H\) is a subgroup of \(G\text{.}\)
43.
View Source for exercise
<exercise number="43">
<statement>
<p>
Prove or disprove: <m>SL_2( {\mathbb Z} )</m>,
the set of <m>2 \times 2</m> matrices with integer entries and determinant one,
is a subgroup of <m>SL_2( {\mathbb R} )</m>.
</p>
</statement>
</exercise>
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.
View Source for exercise
<exercise number="44">
<statement>
<p>
List the subgroups of the quaternion group, <m>Q_8</m>.
</p>
</statement>
</exercise>
List the subgroups of the quaternion group, \(Q_8\text{.}\)
45.
View Source for exercise
<exercise number="45">
<statement>
<p>
Prove that the intersection of two subgroups of a group <m>G</m> is also a subgroup of <m>G</m>.
</p>
</statement>
</exercise>
Prove that the intersection of two subgroups of a group \(G\) is also a subgroup of \(G\text{.}\)
46.
View Source for exercise
<exercise number="46">
<statement>
<p>
Prove or disprove: If <m>H</m> and <m>K</m> are subgroups of a group <m>G</m>,
then <m>H \cup K</m> is a subgroup of <m>G</m>.
</p>
</statement>
<hint>
<p>
Look at <m>S_3</m>.
</p>
</hint>
</exercise>
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.
View Source for hint
<hint>
<p>
Look at <m>S_3</m>.
</p>
</hint>
Look at \(S_3\text{.}\)
47.
View Source for exercise
<exercise number="47">
<statement>
<p>
Prove or disprove: If <m>H</m> and <m>K</m> are subgroups of a group <m>G</m>,
then <m>H K = \{hk : h \in H \text{ and } k \in K \}</m> is a subgroup of <m>G</m>.
What if <m>G</m> is abelian?
</p>
</statement>
</exercise>
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.
View Source for exercise
<exercise number="48">
<statement>
<p>
Let <m>G</m> be a group and <m>g \in G</m>.
Show that
<me>
Z(G) = \{ x \in G : gx = xg \text{ for all } g \in G \}
</me>
is a subgroup of <m>G</m>.
This subgroup is called the <term>center</term> of <m>G</m>.
<notation>
<usage><m>Z(G)</m></usage>
<description>the center of a group</description>
</notation>
</p>
</statement>
</exercise>
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.
View Source for exercise
<exercise number="49">
<statement>
<p>
Let <m>a</m> and <m>b</m> be elements of a group <m>G</m>.
If <m>a^4b = ba</m> and <m>a^3 = e</m>,
prove that <m>ab = ba</m>.
</p>
</statement>
<hint>
<p>
Since <m>a^4b = ba</m>,
it must be the case that <m>b = a^6 b = a^2 b a</m>,
and we can conclude that <m> ab = a^3 b a = ba</m>.
</p>
</hint>
</exercise>
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.
View Source for hint
<hint>
<p>
Since <m>a^4b = ba</m>,
it must be the case that <m>b = a^6 b = a^2 b a</m>,
and we can conclude that <m> ab = a^3 b a = ba</m>.
</p>
</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.
View Source for exercise
<exercise number="50">
<statement>
<p>
Give an example of an infinite group in which every nontrivial subgroup is infinite.
</p>
</statement>
</exercise>
Give an example of an infinite group in which every nontrivial subgroup is infinite.
51.
View Source for exercise
<exercise number="51">
<statement>
<p>
If <m>xy = x^{-1} y^{-1}</m> for all <m>x</m> and <m>y</m> in <m>G</m>,
prove that <m>G</m> must be abelian.
</p>
</statement>
</exercise>
If \(xy = x^{-1} y^{-1}\) for all \(x\) and \(y\) in \(G\text{,}\) prove that \(G\) must be abelian.
52.
View Source for exercise
<exercise number="52">
<statement>
<p>
Prove or disprove: Every proper subgroup of an nonabelian group is nonabelian.
</p>
</statement>
</exercise>
Prove or disprove: Every proper subgroup of an nonabelian group is nonabelian.
53.
View Source for exercise
<exercise number="53">
<statement>
<p>
Let <m>H</m> be a subgroup of <m>G</m> and
<me>
C(H) = \{ g \in G : gh = hg \text{ for all } h \in H \}
</me>.
Prove <m>C(H)</m> is a subgroup of <m>G</m>.
This subgroup is called the <term>centralizer</term> of <m>H</m> in <m>G</m>.
</p>
</statement>
</exercise>
Let \(H\) be a subgroup of \(G\) and
\begin{equation*}
C(H) = \{ g \in G : gh = hg \text{ for all } h \in H \}\text{.}
\end{equation*}
Prove \(C(H)\) is a subgroup of \(G\text{.}\) This subgroup is called the centralizer of \(H\) in \(G\text{.}\)
54.
View Source for exercise
<exercise number="54">
<statement>
<p>
Let <m>H</m> be a subgroup of <m>G</m>.
If <m>g \in G</m>, show that
<m>gHg^{-1} = \{g^{-1}hg : h\in H\}</m> is also a subgroup of <m>G</m>.
</p>
</statement>
</exercise>
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.
View Source for exercisegroup
<exercisegroup cols="2">
<introduction>
<p>
In each group, how many solutions are there to <m>x^2=e</m>?
</p>
</introduction>
<exercise number="55">
<statement>
<p>
<m>C_n</m>, <m>n</m> odd.
</p>
</statement>
<answer>
<p>
<m>1</m>
</p>
</answer>
</exercise>
<exercise number="56">
<statement>
<p>
<m>C_n</m>, <m>n</m> even.
</p>
</statement>
<answer>
<p>
<m>2</m>
</p>
</answer>
</exercise>
<exercise number="57">
<statement>
<p>
<m>D_n</m>, <m>n</m> odd.
</p>
</statement>
<answer>
<p>
<m>n</m>
</p>
</answer>
</exercise>
<exercise number="58">
<statement>
<p>
<m>D_n</m>, <m>n</m> even.
</p>
</statement>
<answer>
<p>
<m>n+1</m>
</p>
</answer>
</exercise>
</exercisegroup>
In each group, how many solutions are there to \(x^2=e\text{?}\)
55.
View Source for exercise
<exercise number="55">
<statement>
<p>
<m>C_n</m>, <m>n</m> odd.
</p>
</statement>
<answer>
<p>
<m>1</m>
</p>
</answer>
</exercise>
\(C_n\text{,}\) \(n\) odd.
Answer.
View Source for answer
<answer>
<p>
<m>1</m>
</p>
</answer>
\(1\)
56.
View Source for exercise
<exercise number="56">
<statement>
<p>
<m>C_n</m>, <m>n</m> even.
</p>
</statement>
<answer>
<p>
<m>2</m>
</p>
</answer>
</exercise>
\(C_n\text{,}\) \(n\) even.
Answer.
View Source for answer
<answer>
<p>
<m>2</m>
</p>
</answer>
\(2\)
57.
View Source for exercise
<exercise number="57">
<statement>
<p>
<m>D_n</m>, <m>n</m> odd.
</p>
</statement>
<answer>
<p>
<m>n</m>
</p>
</answer>
</exercise>
\(D_n\text{,}\) \(n\) odd.
Answer.
View Source for answer
<answer>
<p>
<m>n</m>
</p>
</answer>
\(n\)
58.
View Source for exercise
<exercise number="58">
<statement>
<p>
<m>D_n</m>, <m>n</m> even.
</p>
</statement>
<answer>
<p>
<m>n+1</m>
</p>
</answer>
</exercise>
\(D_n\text{,}\) \(n\) even.
Answer.
View Source for answer
<answer>
<p>
<m>n+1</m>
</p>
</answer>
\(n+1\)
59.
View Source for exercise
<exercise number="59">
<introduction>
<p>
This is an odd-numbered exercise with tasks.
</p>
</introduction>
<task>
<statement>
<p>
What is <m>1+1</m>?
</p>
</statement>
<answer>
<p>
<m>2</m>
</p>
</answer>
</task>
<task>
<introduction>
<p>
This task has subtasks.
</p>
</introduction>
<task>
<statement>
<p>
What is <m>3+3</m>?
</p>
</statement>
<answer>
<p>
<m>6</m>
</p>
</answer>
</task>
<task>
<statement>
<p>
What is <m>5+5</m>?
</p>
</statement>
<answer>
<p>
<m>10</m>
</p>
</answer>
</task>
</task>
</exercise>
This is an odd-numbered exercise with tasks.
(a)
View Source for task
<task>
<statement>
<p>
What is <m>1+1</m>?
</p>
</statement>
<answer>
<p>
<m>2</m>
</p>
</answer>
</task>
What is \(1+1\text{?}\)
Answer.
View Source for answer
<answer>
<p>
<m>2</m>
</p>
</answer>
\(2\)
(b)
View Source for task
<task>
<introduction>
<p>
This task has subtasks.
</p>
</introduction>
<task>
<statement>
<p>
What is <m>3+3</m>?
</p>
</statement>
<answer>
<p>
<m>6</m>
</p>
</answer>
</task>
<task>
<statement>
<p>
What is <m>5+5</m>?
</p>
</statement>
<answer>
<p>
<m>10</m>
</p>
</answer>
</task>
</task>
This task has subtasks.
(i)
View Source for task
<task>
<statement>
<p>
What is <m>3+3</m>?
</p>
</statement>
<answer>
<p>
<m>6</m>
</p>
</answer>
</task>
What is \(3+3\text{?}\)
Answer.
View Source for answer
<answer>
<p>
<m>6</m>
</p>
</answer>
\(6\)
(ii)
View Source for task
<task>
<statement>
<p>
What is <m>5+5</m>?
</p>
</statement>
<answer>
<p>
<m>10</m>
</p>
</answer>
</task>
What is \(5+5\text{?}\)
Answer.
View Source for answer
<answer>
<p>
<m>10</m>
</p>
</answer>
\(10\)
60.
View Source for exercise
<exercise number="60">
<introduction>
<p>
This is an even-numbered exercise with tasks.
</p>
</introduction>
<task>
<statement>
<p>
What is <m>2+2</m>?
</p>
</statement>
<answer>
<p>
<m>4</m>
</p>
</answer>
</task>
<task>
<introduction>
<p>
This task has subtasks.
</p>
</introduction>
<task>
<statement>
<p>
What is <m>4+4</m>?
</p>
</statement>
<answer>
<p>
<m>8</m>
</p>
</answer>
</task>
<task>
<statement>
<p>
What is <m>6+6</m>?
</p>
</statement>
<answer>
<p>
<m>12</m>
</p>
</answer>
</task>
</task>
</exercise>
This is an even-numbered exercise with tasks.
(a)
View Source for task
<task>
<statement>
<p>
What is <m>2+2</m>?
</p>
</statement>
<answer>
<p>
<m>4</m>
</p>
</answer>
</task>
What is \(2+2\text{?}\)
Answer.
View Source for answer
<answer>
<p>
<m>4</m>
</p>
</answer>
\(4\)
(b)
View Source for task
<task>
<introduction>
<p>
This task has subtasks.
</p>
</introduction>
<task>
<statement>
<p>
What is <m>4+4</m>?
</p>
</statement>
<answer>
<p>
<m>8</m>
</p>
</answer>
</task>
<task>
<statement>
<p>
What is <m>6+6</m>?
</p>
</statement>
<answer>
<p>
<m>12</m>
</p>
</answer>
</task>
</task>
This task has subtasks.
(i)
View Source for task
<task>
<statement>
<p>
What is <m>4+4</m>?
</p>
</statement>
<answer>
<p>
<m>8</m>
</p>
</answer>
</task>
What is \(4+4\text{?}\)
Answer.
View Source for answer
<answer>
<p>
<m>8</m>
</p>
</answer>
\(8\)
(ii)
View Source for task
<task>
<statement>
<p>
What is <m>6+6</m>?
</p>
</statement>
<answer>
<p>
<m>12</m>
</p>
</answer>
</task>
What is \(6+6\text{?}\)
Answer.
View Source for answer
<answer>
<p>
<m>12</m>
</p>
</answer>
\(12\)
