Cyclic Groups: exercises

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

1. Prove or disprove each of the following statements.

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

   2. \(U(8)\) is cyclic.

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

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

   5. A group with a finite number of subgroups is finite.

   Gosi nzaghachi

   Hint:

   (a) False; (c) false; (e) true.

2. Find the order of each of the following elements.

   1. \(5 \in {\mathbb Z}_{12}\)

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

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

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

   5. \(72 \in {\mathbb Z}_{240}\)

   6. \(312 \in {\mathbb Z}_{471}\)

   Gosi nzaghachi

   Hint:

   (a) \(12\); (c) infinite; (e) \(10\).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   Gosi nzaghachi

   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) \(\{ 1, -1, i, -i \}\).

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

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

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

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

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

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

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

   Gosi nzaghachi

   Hint:

   (a) \[\begin{aligned}\end{aligned}\].

   (c) \[\begin{aligned}\end{aligned}\].

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

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

7. What are all of the cyclic subgroups of the quaternion group, \(Q_8\)?

8. List all of the cyclic subgroups of \(U(30)\).

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

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

    1. \({\mathbb Z}\)

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

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

    Gosi nzaghachi

    Hint:

    (a) \(0\); (b) \(1, -1\).

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

    Gosi nzaghachi

    Hint:

    \(1, 2, 3, 4, 6, 8, 12, 24\).

12. Find a cyclic group with exactly one generator. Can you find cyclic groups with exactly two generators? Four generators? How about \(n\) generators?

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?

14. Let \[\begin{aligned}\end{aligned}\] be elements in \(GL_2( {\mathbb R} )\). Show that \(A\) and \(B\) have finite orders but \(AB\) does not.

15. Evaluate each of the following.

    1. \((3-2i)+ (5i-6)\)

    2. \((4-5i)-\overline{(4i -4)}\)

    3. \((5-4i)(7+2i)\)

    4. \((9-i) \overline{(9-i)}\)

    5. \(i^{45}\)

    6. \((1+i)+\overline{(1+i)}\)

    Gosi nzaghachi

    Hint:

    (a) \(-3 + 3i\); (c) \(43- 18i\); (e) \(i\)

16. Convert the following complex numbers to the form \(a + bi\).

    1. \(2 \cis(\pi / 6 )\)

    2. \(5 \cis(9\pi/4)\)

    3. \(3 \cis(\pi)\)

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

    Gosi nzaghachi

    Hint:

    (a) \(\sqrt{3} + i\); (c) \(-3\).

17. Change the following complex numbers to polar representation.

    1. \(1-i\)

    2. \(-5\)

    3. \(2+2i\)

    4. \(\sqrt{3} + i\)

    5. \(-3i\)

    6. \(2i + 2 \sqrt{3}\)

    Gosi nzaghachi

    Hint:

    (a) \(\sqrt{2} \cis( 7 \pi /4)\); (c) \(2 \sqrt{2} \cis( \pi /4)\); (e) \(3 \cis(3 \pi/2)\).

18. Calculate each of the following expressions.

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

    2. \((1 - i)^{6}\)

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

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

    5. \(((1-i)/2)^{4}\)

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

    7. \((-2 + 2i)^{-5}\)

    Gosi nzaghachi

    Hint:

    (a) \((1 - i)/2\); (c) \(16(i - \sqrt{3}\, )\); (e) \(-1/4\).

19. Prove each of the following statements.

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

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

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

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

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

    6. \(|z w| = |z| |w|\)

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

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

22. Calculate each of the following.

    1. \(292^{3171} \pmod{ 582}\)

    2. \(2557^{ 341} \pmod{ 5681}\)

    3. \(2071^{ 9521} \pmod{ 4724}\)

    4. \(971^{ 321} \pmod{ 765}\)

    Gosi nzaghachi

    Hint:

    (a) \(292\); (c) \(1523\).

23. Let \(a, b \in G\). Prove the following statements.

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

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

    3. The order of \(ab\) is the same as the order of \(ba\).

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

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

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

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

    Gosi nzaghachi

    Hint:

    \(|\langle g \rangle \cap \langle h \rangle| = 1\).

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

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

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 \(\langle a \rangle \cap \langle b \rangle = \{ e \}\).

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 \(G\).

    Gosi nzaghachi

    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 \(G\).

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 \(G\).

33. 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?

34. 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.

35. Prove that the subgroups of \(\mathbb Z\) are exactly \(n{\mathbb Z}\) for \(n = 0, 1, 2, \ldots\).

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

37. Prove that if \(G\) has no proper nontrivial subgroups, then \(G\) is a cyclic group.

    Gosi nzaghachi

    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 \(G\).

38. Prove that the order of an element in a cyclic group \(G\) must divide the order of the group.

39. Prove that if \(G\) is a cyclic group of order \(m\) and \(d \mid m\), then \(G\) must have a subgroup of order \(d\).

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

Symbols used here

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

\pi

pi
Ratio of a circle's circumference to its diameter, 3.14159…

A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}

matrix
A rectangular array of numbers; a linear map.

x \in A,\ A \subseteq B

element of, subset
x belongs to A; every element of A is in B.

i

imaginary unit
i² = −1.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}

number sets
Naturals, integers, rationals, reals, complex numbers.

\blacksquare\ \text{or}\ \square

end of proof (halmos)
Marks the point where the statement has been established.

a \equiv b \pmod n

congruent modulo n
n divides a − b; a and b have the same remainder.

a \mid b,\ \gcd(a,b)

divides, greatest common divisor
b is a multiple of a; the largest number dividing both.

(G, \cdot),\ e,\ g^{-1}

group, identity, inverse
A set with an operation; the do-nothing element; the element that undoes g.

G \cong H,\ G / N

isomorphic, quotient group
Same structure; the group of cosets of a normal subgroup N.

\mathbb{Z}/n\mathbb{Z},\ \mathbb{Z}_n

integers modulo n
The remainders 0…n−1 with clock arithmetic.

\operatorname{Hom}(A, B),\ f \circ g

arrows from A to B, composition
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.

Jiri gị onwe gị

⌨ Type ✎ Draw

Nsogbu igodo gị

Wepụ

x²√÷×π( )=∫d/dxlim[ ]sinlogeˣ

Parts of this page are adapted from Judson, Abstract Algebra: Theory and Applications (GFDL 1.3). Condensed and re-explained here; errors are ours.

Oge Abstract Algebra

GroupsSubgroups, cosets and Lagrange's theoremCyclic groups and permutation groupsHomomorphisms, normal subgroups and quotient groupsRings and fieldsGalois theory: why the quintic has no formula