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Section 4.4 Exercises

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1

Prove or disprove each of the following statements.

All of the generators of \({\mathbb Z}_{60}\) are prime.

\(U(8)\) is cyclic.

\({\mathbb Q}\) is cyclic.

If every proper subgroup of a group \(G\) is cyclic, then \(G\) is a cyclic group.

A group with a finite number of subgroups is finite.

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2

Find the order of each of the following elements.

\(5 \in {\mathbb Z}_{12}\)

\(\sqrt{3} \in {\mathbb R}\)

\(\sqrt{3} \in {\mathbb R}^\ast\)

\(-i \in {\mathbb C}^\ast\)

\(72 \in {\mathbb Z}_{240}\)

\(312 \in {\mathbb Z}_{471}\)

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3

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

The subgroup of \({\mathbb Z}\) generated by \(7\)

The subgroup of \({\mathbb Z}_{24}\) generated by \(15\)

All subgroups of \({\mathbb Z}_{12}\)

All subgroups of \({\mathbb Z}_{60}\)

All subgroups of \({\mathbb Z}_{13}\)

All subgroups of \({\mathbb Z}_{48}\)

The subgroup generated by 3 in \(U(20)\)

The subgroup generated by 5 in \(U(18)\)

The subgroup of \({\mathbb R}^\ast\) generated by \(7\)

The subgroup of \({\mathbb C}^\ast\) generated by \(i\) where \(i^2 = -1\)

The subgroup of \({\mathbb C}^\ast\) generated by \(2i\)

The subgroup of \({\mathbb C}^\ast\) generated by \((1 + i) / \sqrt{2}\)

The subgroup of \({\mathbb C}^\ast\) generated by \((1 + \sqrt{3}\, i) / 2\)

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4

Find the subgroups of \(GL_2( {\mathbb R })\) generated by each of the following matrices.

\(\displaystyle \begin{pmatrix}
0 & 1 \\
-1 & 0
\end{pmatrix}\)

\(\displaystyle
\begin{pmatrix}
0 & 1/3 \\
3 & 0
\end{pmatrix}\)

\(\displaystyle
\begin{pmatrix}
1 & -1 \\
1 & 0
\end{pmatrix}\)

\(\displaystyle
\begin{pmatrix}
1 & -1 \\
0 & 1
\end{pmatrix}\)

\(\displaystyle
\begin{pmatrix}
1 & -1 \\
-1 & 0
\end{pmatrix}\)

\(\displaystyle
\begin{pmatrix}
\sqrt{3}/ 2 & 1/2 \\
-1/2 & \sqrt{3}/2
\end{pmatrix}\)

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5

Find the order of every element in \({\mathbb Z}_{18}\text{.}\)

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6

Find the order of every element in the symmetry group of the square, \(D_4\text{.}\)

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7

What are all of the cyclic subgroups of the quaternion group, \(Q_8\text{?}\)

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8

List all of the cyclic subgroups of \(U(30)\text{.}\)

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9

List every generator of each subgroup of order 8 in \({\mathbb Z}_{32}\text{.}\)

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10

Find all elements of finite order in each of the following groups. Here the “\(\ast\)” indicates the set with zero removed.

\({\mathbb Z}\)

\({\mathbb Q}^\ast\)

\({\mathbb R}^\ast\)

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11

If \(a^{24} =e\) in a group \(G\text{,}\) what are the possible orders of \(a\text{?}\)

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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 \(n \leq 20\text{,}\) 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

\begin{equation*}
A =
\begin{pmatrix}
0 & 1 \\
-1 & 0
\end{pmatrix}
\qquad \text{and} \qquad
B =
\begin{pmatrix}
0 & -1 \\
1 & -1
\end{pmatrix}
\end{equation*}

be elements in \(GL_2( {\mathbb R} )\text{.}\) Show that \(A\) and \(B\) have finite orders but \(AB\) does not.

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15

Evaluate each of the following.

\((3-2i)+ (5i-6)\)

\((4-5i)-\overline{(4i -4)}\)

\((5-4i)(7+2i)\)

\((9-i) \overline{(9-i)}\)

\(i^{45}\)

\((1+i)+\overline{(1+i)}\)

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16

Convert the following complex numbers to the form \(a + bi\text{.}\)

\(2 \cis(\pi / 6 )\)

\(5 \cis(9\pi/4)\)

\(3 \cis(\pi)\)

\(\cis(7\pi/4) /2\)

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17

Change the following complex numbers to polar representation.

\(1-i\)

\(-5\)

\(2+2i\)

\(\sqrt{3} + i\)

\(-3i\)

\(2i + 2 \sqrt{3}\)

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18

Calculate each of the following expressions.

\((1+i)^{-1}\)

\((1 - i)^{6}\)

\((\sqrt{3} + i)^{5}\)

\((-i)^{10}\)

\(((1-i)/2)^{4}\)

\((-\sqrt{2} - \sqrt{2}\, i)^{12}\)

\((-2 + 2i)^{-5}\)

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19

Prove each of the following statements.

\(|z| = | \overline{z}|\)

\(z \overline{z} = |z|^2\)

\(z^{-1} = \overline{z} / |z|^2\)

\(|z +w| \leq |z| + |w|\)

\(|z - w| \geq | |z| - |w||\)

\(|z w| = |z| |w|\)

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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.

\(292^{3171} \pmod{ 582}\)

\(2557^{ 341} \pmod{ 5681}\)

\(2071^{ 9521} \pmod{ 4724}\)

\(971^{ 321} \pmod{ 765}\)

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23

Let \(a, b \in G\text{.}\) Prove the following statements.

The order of \(a\) is the same as the order of \(a^{-1}\text{.}\)

For all \(g \in G\text{,}\) \(|a| = |g^{-1}ag|\text{.}\)

The order of \(ab\) is the same as the order of \(ba\text{.}\)

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24

Let \(p\) and \(q\) be distinct primes. How many generators does \({\mathbb Z}_{pq}\) have?

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25

Let \(p\) be prime and \(r\) be a positive integer. How many generators does \({\mathbb Z}_{p^r}\) have?

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26

Prove that \({\mathbb Z}_{p}\) has no nontrivial subgroups if \(p\) is prime.

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27

If \(g\) and \(h\) have orders \(15\) and \(16\) respectively in a group \(G\text{,}\) what is the order of \(\langle g \rangle \cap \langle h \rangle \text{?}\)

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28

Let \(a\) be an element in a group \(G\text{.}\) What is a generator for the subgroup \(\langle a^m \rangle \cap \langle a^n \rangle\text{?}\)

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29

Prove that \({\mathbb Z}_n\) has an even number of generators for \(n \gt 2\text{.}\)

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30

Suppose that \(G\) is a group and let \(a\text{,}\) \(b \in G\text{.}\) Prove that if \(|a| = m\) and \(|b| = n\) with \(\gcd(m,n) = 1\text{,}\) then \(\langle a \rangle \cap \langle b \rangle = \{ e \}\text{.}\)

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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 \(G\text{.}\)

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32

Let \(G\) be a finite cyclic group of order \(n\) generated by \(x\text{.}\) Show that if \(y = x^k\) where \(\gcd(k,n) = 1\text{,}\) then \(y\) must be a generator of \(G\text{.}\)

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33

If \(G\) is an abelian group that contains a pair of cyclic subgroups of order \(2\text{,}\) show that \(G\) must contain a subgroup of order \(4\text{.}\) Does this subgroup have to be cyclic?

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34

Let \(G\) be an abelian group of order \(pq\) where \(\gcd(p,q) = 1\text{.}\) If \(G\) 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 \(\mathbb Z\) are exactly \(n{\mathbb Z}\) for \(n = 0, 1, 2, \ldots\text{.}\)

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36

Prove that the generators of \({\mathbb Z}_n\) are the integers \(r\) such that \(1 \leq r \lt n\) and \(\gcd(r,n) = 1\text{.}\)

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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 \(d \mid m\text{,}\) then \(G\) must have a subgroup of order \(d\text{.}\)

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40

For what integers \(n\) is \(-1\) an \(n\)th root of unity?

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41

If \(z = r( \cos \theta + i \sin \theta)\) and \(w = s(\cos \phi + i \sin \phi)\) are two nonzero complex numbers, show that

\begin{equation*}
zw = rs[ \cos( \theta + \phi) + i \sin( \theta + \phi)].
\end{equation*}

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42

Prove that the circle group is a subgroup of \({\mathbb C}^*\text{.}\)

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43

Prove that the \(n\)th roots of unity form a cyclic subgroup of \({\mathbb T}\) of order \(n\text{.}\)

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44

Let \(\alpha \in \mathbb T\text{.}\) Prove that \(\alpha^m =1\) and \(\alpha^n = 1\) if and only if \(\alpha^d = 1\) for \(d = \gcd(m,n)\text{.}\)

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45

Let \(z \in {\mathbb C}^\ast\text{.}\) If \(|z| \neq 1\text{,}\) prove that the order of \(z\) is infinite.

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46

Let \(z =\cos \theta + i \sin \theta\) be in \({\mathbb T}\) where \(\theta \in {\mathbb Q}\text{.}\) Prove that the order of \(z\) is infinite.