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Cyclic Groups: exercises
Cyclic Groups: exercises — from Judson, Abstract Algebra: Theory and Applications.
Practice (40)
Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.
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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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Hint:
(a) False; (c) false; (e) true.
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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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Hint:
(a) \(12\); (c) infinite; (e) \(10\).
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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 \((1 + \sqrt{3}\, i) / 2\)
Revelar la respuesta
Hint:
(a) \(7 {\mathbb Z} = \{ \ldots, -7, 0, 7, 14, \ldots \}\); (b) \(\{ 0, 3, 6, 9, 12, 15, 18, 21 \}\); (c) \(\{ 0 \}\), \(\{ 0, 6 \}\), \(\{ 0, 4, 8 \}\), \(\{ 0, 3, 6, 9 \}\), \(\{ 0, 2, 4, 6, 8, 10 \}\); (g) \(\{ 1, 3, 7, 9 \}\); (j) \(\{ 1, -1, i, -i \}\).
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Find the subgroups of \(GL_2( {\mathbb R })\) generated by each of the following matrices.
\(\displaystyle \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}\)
\(\displaystyle \begin{pmatrix} 0 & 1/3 \\ 3 & 0 \end{pmatrix}\)
\(\displaystyle \begin{pmatrix} 1 & -1 \\ 1 & 0 \end{pmatrix}\)
\(\displaystyle \begin{pmatrix} 1 & -1 \\ 0 & 1 \end{pmatrix}\)
\(\displaystyle \begin{pmatrix} 1 & -1 \\ -1 & 0 \end{pmatrix}\)
\(\displaystyle \begin{pmatrix} \sqrt{3}/ 2 & 1/2 \\ -1/2 & \sqrt{3}/2 \end{pmatrix}\)
Revelar la respuesta
Hint:
(a) \[\begin{aligned}\end{aligned}\].
(c) \[\begin{aligned}\end{aligned}\].
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Find the order of every element in \({\mathbb Z}_{18}\).
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Find the order of every element in the symmetry group of the square, \(D_4\).
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What are all of the cyclic subgroups of the quaternion group, \(Q_8\)?
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List all of the cyclic subgroups of \(U(30)\).
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List every generator of each subgroup of order 8 in \({\mathbb Z}_{32}\).
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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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Hint:
(a) \(0\); (b) \(1, -1\).
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If \(a^{24} =e\) in a group \(G\), what are the possible orders of \(a\)?
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Hint:
\(1, 2, 3, 4, 6, 8, 12, 24\).
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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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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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Let \[\begin{aligned}\end{aligned}\] be elements in \(GL_2( {\mathbb R} )\). Show that \(A\) and \(B\) have finite orders but \(AB\) does not.
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Evaluate each of the following.
\((3-2i)+ (5i-6)\)
\((4-5i)-\overline{(4i -4)}\)
\((5-4i)(7+2i)\)
\((9-i) \overline{(9-i)}\)
\(i^{45}\)
\((1+i)+\overline{(1+i)}\)
Revelar la respuesta
Hint:
(a) \(-3 + 3i\); (c) \(43- 18i\); (e) \(i\)
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Convert the following complex numbers to the form \(a + bi\).
\(2 \cis(\pi / 6 )\)
\(5 \cis(9\pi/4)\)
\(3 \cis(\pi)\)
\(\cis(7\pi/4) /2\)
Revelar la respuesta
Hint:
(a) \(\sqrt{3} + i\); (c) \(-3\).
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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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Hint:
(a) \(\sqrt{2} \cis( 7 \pi /4)\); (c) \(2 \sqrt{2} \cis( \pi /4)\); (e) \(3 \cis(3 \pi/2)\).
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Calculate each of the following expressions.
\((1+i)^{-1}\)
\((1 - i)^{6}\)
\((\sqrt{3} + i)^{5}\)
\((-i)^{10}\)
\(((1-i)/2)^{4}\)
\((-\sqrt{2} - \sqrt{2}\, i)^{12}\)
\((-2 + 2i)^{-5}\)
Revelar la respuesta
Hint:
(a) \((1 - i)/2\); (c) \(16(i - \sqrt{3}\, )\); (e) \(-1/4\).
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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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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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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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Calculate each of the following.
\(292^{3171} \pmod{ 582}\)
\(2557^{ 341} \pmod{ 5681}\)
\(2071^{ 9521} \pmod{ 4724}\)
\(971^{ 321} \pmod{ 765}\)
Revelar la respuesta
Hint:
(a) \(292\); (c) \(1523\).
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Let \(a, b \in G\). Prove the following statements.
The order of \(a\) is the same as the order of \(a^{-1}\).
For all \(g \in G\), \(|a| = |g^{-1}ag|\).
The order of \(ab\) is the same as the order of \(ba\).
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Let \(p\) and \(q\) be distinct primes. How many generators does \({\mathbb Z}_{pq}\) have?
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Let \(p\) be prime and \(r\) be a positive integer. How many generators does \({\mathbb Z}_{p^r}\) have?
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Prove that \({\mathbb Z}_{p}\) has no nontrivial subgroups if \(p\) is prime.
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If \(g\) and \(h\) have orders \(15\) and \(16\) respectively in a group \(G\), what is the order of \(\langle g \rangle \cap \langle h \rangle\)?
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Hint:
\(|\langle g \rangle \cap \langle h \rangle| = 1\).
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Let \(a\) be an element in a group \(G\). What is a generator for the subgroup \(\langle a^m \rangle \cap \langle a^n \rangle\)?
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Prove that \({\mathbb Z}_n\) has an even number of generators for \(n \gt 2\).
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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 \(\langle a \rangle \cap \langle b \rangle = \{ e \}\).
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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 \(G\).
Revelar la respuesta
Hint:
The identity element in any group has finite order. Let \(g, h \in G\) have orders \(m\) and \(n\), respectively. Since \((g^{-1})^m = e\) and \((gh)^{mn} = e\), the elements of finite order in \(G\) form a subgroup of \(G\).
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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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If \(G\) 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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Let \(G\) be an abelian group of order \(pq\) where \(\gcd(p,q) = 1\). If \(G\) contains elements \(a\) and \(b\) of order \(p\) and \(q\) respectively, then show that \(G\) is cyclic.
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Prove that the subgroups of \(\mathbb Z\) are exactly \(n{\mathbb Z}\) for \(n = 0, 1, 2, \ldots\).
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Prove that the generators of \({\mathbb Z}_n\) are the integers \(r\) such that \(1 \leq r \lt n\) and \(\gcd(r,n) = 1\).
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Prove that if \(G\) has no proper nontrivial subgroups, then \(G\) is a cyclic group.
Revelar la respuesta
Hint:
If \(g\) is an element distinct from the identity in \(G\), \(g\) must generate \(G\); otherwise, \(\langle g \rangle\) is a nontrivial proper subgroup of \(G\).
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Prove that the order of an element in a cyclic group \(G\) must divide the order of the group.
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Prove that if \(G\) is a cyclic group of order \(m\) and \(d \mid m\), then \(G\) must have a subgroup of order \(d\).
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For what integers \(n\) is \(-1\) an \(n\)th root of unity?
Symbols used here
The non-negative number whose square (n-th power) is x.
Ratio of a circle's circumference to its diameter, 3.14159…
A rectangular array of numbers; a linear map.
x belongs to A; every element of A is in B.
i² = −1.
Inequalities that allow equality; < and > exclude it.
Naturals, integers, rationals, reals, complex numbers.
Marks the point where the statement has been established.
n divides a − b; a and b have the same remainder.
b is a multiple of a; the largest number dividing both.
A set with an operation; the do-nothing element; the element that undoes g.
Same structure; the group of cosets of a normal subgroup N.
The remainders 0…n−1 with clock arithmetic.
The set of morphisms; do g then f.
Questions people ask
What is a group, in plain words?
A set with one operation that is associative, has an identity, and lets every element be undone. Symmetries of any object form a group — that is where the idea came from.
What is the difference between a ring and a field?
A ring has addition and multiplication that behave like the integers (you cannot always divide); a field is a ring where every non-zero element has a reciprocal, like the rationals or the reals.
Prueba tu propio
Parts of this page are adapted from Judson, Abstract Algebra: Theory and Applications (GFDL 1.3). Condensed and re-explained here; errors are ours.
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