SB Exercises

Exercises 4.5 Exercises

1.

Prove or disprove each of the following statements.

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All of the generators of $Z_{60}$ are prime.
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$U (8)$ is cyclic.
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$Q$ is cyclic.
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If every proper subgroup of a group $G$ is cyclic, then $G$ is a cyclic group.
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A group with a finite number of subgroups is finite.
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2.

Find the order of each of the following elements.

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$5 \in Z_{12}$
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$\sqrt{3} \in R$
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$\sqrt{3} \in R^{*}$
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$- i \in C^{*}$
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72 in $Z_{240}$
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312 in $Z_{471}$
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3.

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

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The subgroup of $Z$ generated by 7
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The subgroup of $Z_{24}$ generated by 15
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All subgroups of $Z_{12}$
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All subgroups of $Z_{60}$
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All subgroups of $Z_{13}$
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All subgroups of $Z_{48}$
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The subgroup generated by 3 in $U (20)$
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The subgroup generated by 5 in $U (18)$
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The subgroup of $R^{*}$ generated by 7
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The subgroup of $C^{*}$ generated by $i$ where $i^{2} = - 1$
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The subgroup of $C^{*}$ generated by $2 i$
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The subgroup of $C^{*}$ generated by $(1 + i) / \sqrt{2}$
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The subgroup of $C^{*}$ generated by $(1 + \sqrt{3} i) / 2$
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4.

Find the subgroups of

G L_{2} (R)

generated by each of the following matrices.

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$(\begin{matrix} 0 & 1 \\ - 1 & 0 \end{matrix})$
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$(\begin{matrix} 0 & 1 / 3 \\ 3 & 0 \end{matrix})$
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$(\begin{matrix} 1 & - 1 \\ 1 & 0 \end{matrix})$
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$(\begin{matrix} 1 & - 1 \\ 0 & 1 \end{matrix})$
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$(\begin{matrix} 1 & - 1 \\ - 1 & 0 \end{matrix})$
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$(\begin{matrix} \sqrt{3} / 2 & 1 / 2 \\ - 1 / 2 & \sqrt{3} / 2 \end{matrix})$
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5.

Find the order of every element in

Z_{18} .

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6.

Find the order of every element in the symmetry group of the square,

D_{4} .

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7.

What are all of the cyclic subgroups of the quaternion group,

Q_{8} ?

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8.

List all of the cyclic subgroups of

U (30) .

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9.

List every generator of each subgroup of order 8 in

Z_{32} .

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10.

Find all elements of finite order in each of the following groups. Here the “

*

” indicates the set with zero removed.

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$Z$
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$Q^{*}$
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$R^{*}$
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11.

a^{24} = e

in a group

G,

what are the possible orders of

a ?

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12.

Find a cyclic group with exactly one generator. Can you find cyclic groups with exactly two generators? Four generators? How about

n

generators?

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13.

For

n \leq 20,

which groups

U (n)

are cyclic? Make a conjecture as to what is true in general. Can you prove your conjecture?

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14.

Let

A = (\begin{matrix} 0 & 1 \\ - 1 & 0 \end{matrix}) and B = (\begin{matrix} 0 & - 1 \\ 1 & - 1 \end{matrix})

be elements in

G L_{2} (R) .

Show that

A

and

B

have finite orders but

A B

does not.

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15.

Evaluate each of the following.

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$(3 - 2 i) + (5 i - 6)$
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$(4 - 5 i) - \overset{―}{(4 i - 4)}$
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$(5 - 4 i) (7 + 2 i)$
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$(9 - i) \overset{―}{(9 - i)}$
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$i^{45}$
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$(1 + i) + \overset{―}{(1 + i)}$
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16.

Convert the following complex numbers to the form

a + b i .

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$2 cis (π / 6)$
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$5 cis (9 π / 4)$
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$3 cis (π)$
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$cis (7 π / 4) / 2$
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17.

Change the following complex numbers to polar representation.

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$1 - i$
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$- 5$
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$2 + 2 i$
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$\sqrt{3} + i$
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$- 3 i$
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$2 i + 2 \sqrt{3}$
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18.

Calculate each of the following expressions.

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$(1 + i)^{- 1}$
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$(1 - i)^{6}$
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$(\sqrt{3} + i)^{5}$
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$(- i)^{10}$
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$((1 - i) / 2)^{4}$
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$(- \sqrt{2} - \sqrt{2} i)^{12}$
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$(- 2 + 2 i)^{- 5}$
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19.

Prove each of the following statements.

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$| z | = | \overset{―}{z} |$
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$z \overset{―}{z} = | z |^{2}$
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$z^{- 1} = \overset{―}{z} / | z |^{2}$
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$| z + w | \leq | z | + | w |$
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$| z - w | \geq | | z | - | w | |$
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$| z w | = | z | | w |$
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20.

List and graph the 6th roots of unity. What are the generators of this group? What are the primitive 6th roots of unity?

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21.

List and graph the 5th roots of unity. What are the generators of this group? What are the primitive 5th roots of unity?

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22.

Calculate each of the following.

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$292^{3171} (\mod 582)$
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$2557^{341} (\mod 5681)$
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$2071^{9521} (\mod 4724)$
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$971^{321} (\mod 765)$
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23.

Let

a, b \in G .

Prove the following statements.

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The order of $a$ is the same as the order of $a^{- 1} .$
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For all $g \in G,$ $| a | = | g^{- 1} a g | .$
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The order of $a b$ is the same as the order of $b a .$
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24.

Let

p

and

q

be distinct primes. How many generators does

Z_{p q}

have?

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25.

Let

p

be prime and

r

be a positive integer. How many generators does

Z_{p^{r}}

have?

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26.

Prove that

Z_{p}

has no nontrivial subgroups if

p

is prime.

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27.

g

and

h

have orders 15 and 16 respectively in a group

G,

what is the order of

⟨ g ⟩ \cap ⟨ h ⟩ ?

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28.

Let

a

be an element in a group

G .

What is a generator for the subgroup

⟨ a^{m} ⟩ \cap ⟨ a^{n} ⟩ ?

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29.

Prove that

Z_{n}

has an even number of generators for

n > 2 .

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30.

Suppose that

G

is a group and let

a,

b \in G .

Prove that if

| a | = m

and

| b | = n

with

gcd (m, n) = 1,

then

⟨ a ⟩ \cap ⟨ b ⟩ = {e} .

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31.

Let

G

be an abelian group. Show that the elements of finite order in

G

form a subgroup. This subgroup is called the torsion subgroup of

G .

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32.

Let

G

be a finite cyclic group of order

n

generated by

x .

Show that if

y = x^{k}

where

gcd (k, n) = 1,

then

y

must be a generator of

G .

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33.

G

is an abelian group that contains a pair of cyclic subgroups of order 2, show that

G

must contain a subgroup of order 4. Does this subgroup have to be cyclic?

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34.

Let

G

be an abelian group of order

p q

where

gcd (p, q) = 1 .

G

contains elements

a

and

b

of order

p

and

q

respectively, then show that

G

is cyclic.

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35.

Prove that the subgroups of

Z

are exactly

n Z

for

n = 0, 1, 2, \dots .

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36.

Prove that the generators of

Z_{n}

are the integers

r

such that

1 \leq r < n

and

gcd (r, n) = 1 .

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37.

Prove that if

G

has no proper nontrivial subgroups, then

G

is a cyclic group.

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38.

Prove that the order of an element in a cyclic group

G

must divide the order of the group.

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39.

Prove that if

G

is a cyclic group of order

m

and

d ∣ m,

then

G

must have a subgroup of order

d .

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40.

For what integers

n

- 1

n

th root of unity?

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41.

z = r (\cos θ + i \sin θ)

and

w = s (\cos ϕ + i \sin ϕ)

are two nonzero complex numbers, show that

z w = r s [\cos (θ + ϕ) + i \sin (θ + ϕ)] .

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42.

Prove that the circle group is a subgroup of

C^{*} .

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43.

Prove that the

n

th roots of unity form a cyclic subgroup of

T

of order

n .

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44.

Let

α \in T .

Prove that

α^{m} = 1

and

α^{n} = 1

if and only if

α^{d} = 1

for

d = gcd (m, n) .

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45.

Let

z \in C^{*} .

| z | \neq 1,

prove that the order of

z

is infinite.

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46.

Let

z = \cos θ + i \sin θ

be in

T

where

θ \in Q .

Prove that the order of

z

is infinite.

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