SB Exercises

Exercises 3.5 Exercises

1.

Find all

x \in Z

satisfying each of the following equations.

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$3 x \equiv 2 (\mod 7)$
$5 x + 1 \equiv 13 (\mod 23)$
$5 x + 1 \equiv 13 (\mod 26)$
$9 x \equiv 3 (\mod 5)$
$5 x \equiv 1 (\mod 6)$
$3 x \equiv 1 (\mod 6)$

Hint.

(a)

3 + 7 Z = {\dots, - 4, 3, 10, \dots};

(c)

18 + 26 Z;

(e)

5 + 6 Z .

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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.

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$\begin{array}{ccccc} \circ & a & b & c & d \\ a & a & c & d & a \\ b & b & b & c & d \\ c & c & d & a & b \\ d & d & a & b & c \end{array}$
$\begin{array}{ccccc} \circ & a & b & c & d \\ a & a & b & c & d \\ b & b & a & d & c \\ c & c & d & a & b \\ d & d & c & b & a \end{array}$
$\begin{array}{ccccc} \circ & a & b & c & d \\ a & a & b & c & d \\ b & b & c & d & a \\ c & c & d & a & b \\ d & d & a & b & c \end{array}$
$\begin{array}{ccccc} \circ & a & b & c & d \\ a & a & b & c & d \\ b & b & a & c & d \\ c & c & b & a & d \\ d & d & d & b & c \end{array}$

Hint.

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

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3.

Write out Cayley tables for groups formed by the symmetries of a rectangle and for

(Z_{4}, +) .

How many elements are in each group? Are the groups the same? Why or why not?

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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?

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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} .

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6.

Give a multiplication table for the group

U (12) .

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Hint.

\begin{array}{ccccc} \cdot & 1 & 5 & 7 & 11 \\ 1 & 1 & 5 & 7 & 11 \\ 5 & 5 & 1 & 11 & 7 \\ 7 & 7 & 11 & 1 & 5 \\ 11 & 11 & 7 & 5 & 1 \end{array}

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7.

Let

S = R ∖ {- 1}

and define a binary operation on

S

a * b = a + b + a b .

Prove that

(S, *)

is an abelian group.

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8.

Give an example of two elements

A

and

B

G L_{2} (R)

with

A B \neq B A .

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Hint.

Pick two matrices. Almost any pair will work.

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9.

Prove that the product of two matrices in

S L_{2} (R)

has determinant one.

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10.

Prove that the set of matrices of the form

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(\begin{matrix} 1 & x & y \\ 0 & 1 & z \\ 0 & 0 & 1 \end{matrix})

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

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(\begin{matrix} 1 & x & y \\ 0 & 1 & z \\ 0 & 0 & 1 \end{matrix}) (\begin{matrix} 1 & x^{'} & y^{'} \\ 0 & 1 & z^{'} \\ 0 & 0 & 1 \end{matrix}) = (\begin{matrix} 1 & x + x^{'} & y + y^{'} + x z^{'} \\ 0 & 1 & z + z^{'} \\ 0 & 0 & 1 \end{matrix}) .

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11.

Prove that

det (A B) = det (A) det (B)

G L_{2} (R) .

Use this result to show that the binary operation in the group

G L_{2} (R)

is closed; that is, if

A

and

B

are in

G L_{2} (R),

then

A B \in G L_{2} (R) .

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12.

Let

Z_{2}^{n} = {(a_{1}, a_{2}, \dots, a_{n}) : a_{i} \in Z_{2}} .

Define a binary operation on

Z_{2}^{n}

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(a_{1}, a_{2}, \dots, a_{n}) + (b_{1}, b_{2}, \dots, b_{n}) = (a_{1} + b_{1}, a_{2} + b_{2}, \dots, a_{n} + b_{n}) .

Prove that

Z_{2}^{n}

is a group under this operation. This group is important in algebraic coding theory.

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13.

Show that

R^{*} = R ∖ {0}

is a group under the operation of multiplication.

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14.

Given the groups

R^{*}

and

Z,

let

G = R^{*} \times Z .

Define a binary operation

\circ

G

(a, m) \circ (b, n) = (a b, m + n) .

Show that

G

is a group under this operation.

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15.

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

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Hint.

There is a nonabelian group containing six elements.

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16.

Give a specific example of some group

G

and elements

g, h \in G

where

(g h)^{n} \neq g^{n} h^{n} .

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Hint.

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

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17.

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

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Hint.

The are five different groups of order 8.

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18.

Show that there are

n!

permutations of a set containing

n

items.

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Hint.

Let

σ = (\begin{matrix} 1 & 2 & \dots & n \\ a_{1} & a_{2} & \dots & a_{n} \end{matrix})

be in

S_{n} .

All of the

a_{i}

s must be distinct. There are

n

ways to choose

a_{1},

n - 1

ways to choose

a_{2},

\dots,

2 ways to choose

a_{n - 1},

and only one way to choose

a_{n} .

Therefore, we can form

σ

n (n - 1) \dots 2 \cdot 1 = n!

ways.

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19.

Show that

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0 + a \equiv a + 0 \equiv a (\mod n)

for all

a \in Z_{n} .

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20.

Prove that there is a multiplicative identity for the integers modulo

n :

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a \cdot 1 \equiv a (\mod n) .

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21.

For each

a \in Z_{n}

find an element

b \in Z_{n}

such that

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a + b \equiv b + a \equiv 0 (\mod n) .

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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 .

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23.

Show that addition and multiplication mod

n

are associative operations.

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24.

Show that multiplication distributes over addition modulo

n :

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a (b + c) \equiv a b + a c (\mod n) .

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25.

Let

a

and

b

be elements in a group

G .

Prove that

a b^{n} a^{- 1} = (a b a^{- 1})^{n}

for

n \in Z .

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Hint.

\begin{aligned} (a b a^{- 1})^{n} & = (a b a^{- 1}) (a b a^{- 1}) \dots (a b a^{- 1}) \\ = a b (a a^{- 1}) b (a a^{- 1}) b \dots b (a a^{- 1}) b a^{- 1} \\ = a b^{n} a^{- 1} . \end{aligned}

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26.

Let

U (n)

be the group of units in

Z_{n} .

n > 2,

prove that there is an element

k \in U (n)

such that

k^{2} = 1

and

k \neq 1 .

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27.

Prove that the inverse of

g_{1} g_{2} \dots g_{n}

g_{n}^{- 1} g_{n - 1}^{- 1} \dots g_{1}^{- 1} .

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28.

Prove the remainder of Proposition 3.2.14: if

G

is a group and

a, b \in G,

then the equation

x a = b

has a unique solution in

G .

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29.

Prove Theorem 3.2.16.

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30.

Prove the right and left cancellation laws for a group

G;

that is, show that in the group

G,

b a = c a

implies

b = c

and

a b = a c

implies

b = c

for elements

a, b, c \in G .

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31.

Show that if

a^{2} = e

for all elements

a

in a group

G,

then

G

must be abelian.

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Hint.

Since

a b a b = (a b)^{2} = e = a^{2} b^{2} = a a b b,

we know that

b a = a b .

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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 .

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33.

Let

G

be a group and suppose that

(a b)^{2} = a^{2} b^{2}

for all

a

and

b

G .

Prove that

G

is an abelian group.

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34.

Find all the subgroups of

Z_{3} \times Z_{3} .

Use this information to show that

Z_{3} \times Z_{3}

is not the same group as

Z_{9} .

(See Example 3.3.5 for a short description of the product of groups.)

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35.

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

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Hint.

H_{1} = {i d},

H_{2} = {i d, ρ_{1}, ρ_{2}},

H_{3} = {i d, μ_{1}},

H_{4} = {i d, μ_{2}},

H_{5} = {i d, μ_{3}},

S_{3} .

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36.

Compute the subgroups of the symmetry group of a square.

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37.

Let

H = {2^{k} : k \in Z} .

Show that

H

is a subgroup of

Q^{*} .

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38.

Let

n = 0, 1, 2, \dots

and

n Z = {n k : k \in Z} .

Prove that

n Z

is a subgroup of

Z .

Show that these subgroups are the only subgroups of

Z .

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39.

Let

T = {z \in C^{*} : | z | = 1} .

Prove that

T

is a subgroup of

C^{*} .

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40.

(\begin{matrix} \cos θ & - \sin θ \\ \sin θ & \cos θ \end{matrix})

where

θ \in R .

Prove that

G

is a subgroup of

S L_{2} (R) .

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41.

Prove that

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G = {a + b \sqrt{2} : a, b \in Q and a and b are not both zero}

is a subgroup of

R^{*}

under the group operation of multiplication.

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Hint.

The identity of

G

1 = 1 + 0 \sqrt{2} .

Since

(a + b \sqrt{2}) (c + d \sqrt{2}) = (a c + 2 b d) + (a d + b c) \sqrt{2},

G

is closed under multiplication. Finally,

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

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42.

Let

G

be the group of

2 \times 2

matrices under addition and

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H = {(\begin{matrix} a & b \\ c & d \end{matrix}) : a + d = 0} .

Prove that

H

is a subgroup of

G .

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43.

Prove or disprove:

S L_{2} (Z),

the set of

2 \times 2

matrices with integer entries and determinant one, is a subgroup of

S L_{2} (R) .

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44.

List the subgroups of the quaternion group,

Q_{8} .

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45.

Prove that the intersection of two subgroups of a group

G

is also a subgroup of

G .

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46.

Prove or disprove: If

H

and

K

are subgroups of a group

G,

then

H \cup K

is a subgroup of

G .

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Hint.

Look at

S_{3} .

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47.

Prove or disprove: If

H

and

K

are subgroups of a group

G,

then

H K = {h k : h \in H and k \in K}

is a subgroup of

G .

What if

G

is abelian?

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48.

Let

G

be a group and

g \in G .

Show that

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Z (G) = {x \in G : g x = x g for all g \in G}

is a subgroup of

G .

This subgroup is called the center of

G .

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49.

Let

a

and

b

be elements of a group

G .

a^{4} b = b a

and

a^{3} = e,

prove that

a b = b a .

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Hint.

Since

a^{4} b = b a,

it must be the case that

b = a^{6} b = a^{2} b a,

and we can conclude that

a b = a^{3} b a = b a .

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50.

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

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51.

x y = x^{- 1} y^{- 1}

for all

x

and

y

G,

prove that

G

must be abelian.

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52.

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

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53.

Let

H

be a subgroup of

G

and

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C (H) = {g \in G : g h = h g for all h \in H} .

Prove

C (H)

is a subgroup of

G .

This subgroup is called the centralizer of

H

G .

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54.

Let

H

be a subgroup of

G .

g \in G,

show that

g H g^{- 1} = {g^{- 1} h g : h \in H}

is also a subgroup of

G .

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Exercise Group.

In each group, how many solutions are there to

x^{2} = e ?

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55.

C_{n},

n

odd.

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Answer.

1

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56.

C_{n},

n

even.

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Answer.

2

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57.

D_{n},

n

odd.

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Answer.

n

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58.

D_{n},

n

even.

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Answer.

n + 1

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59.

This is an odd-numbered exercise with tasks.

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(a)

What is

1 + 1 ?

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Answer.

2

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(b)

This task has subtasks.

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(i)

What is

3 + 3 ?

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Answer.

6

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(ii)

What is

5 + 5 ?

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Answer.

10

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60.

This is an even-numbered exercise with tasks.

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(a)

What is

2 + 2 ?

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Answer.

4

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(b)

This task has subtasks.

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(i)

What is

4 + 4 ?

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Answer.

8

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(ii)

What is

6 + 6 ?

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Answer.

12

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