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Math 128A, problem set 04

CORRECTED Fri Sep 25

Outline due: Wed Sep 23

Due: Mon Sep 28

Last revision due: Wed Oct 21

Problems to be done, but not turned in: (Ch. 4) 17–75 odd; (Ch. 5) 1–19 odd.

Fun: (Ch. 4) 50, 64, 77.

Problems to be turned in:

1. Justify all answers.

(a) List all generators of the subgroup h5i of Z.

(b) Let G = hai be an infinite cyclic group. List all generators of the subgroup a5

of G. (corrected )

(c) Z60 has a subgroup H of order 20. List all generators of that subgroup H.

(d) Let G = hai be a cyclic subgroup of order 60, and let H be a subgroup of G of

order 20. List all generators of that subgroup H.

2. Find the subgroup lattices for Z5 , Z10 , Z70 , and Z770 . Generalize as much as you can.

3. (Ch. 4) 42. Prove your answer.

4. (Ch. 4) 68. Prove your answer.

5. (a) Let G be an abelian group of order 119 such that x119 = e for all x ∈ G. Prove

that G is cyclic.

(b) Let G be an abelian group of order 49 such that x49 = e for all x ∈ G, and

suppose that G is not cyclic. What can you say about G?

6. Let α = (1 7 4)(2 5 3 9 10 8 6 12) and β = (1 2 9 3 10 5)(6 8)(7 12 11) be elements

of S12 .

(a) Compute αβ, in cycle form.

(b) Find the orders of α, β, and αβ.

7. The cycle shape of α ∈ Sn is the set (or actually, multiset) of the lengths of the cycles

obtained when α is expressed as a product of disjoint cycles (see pp. 102–103).

(a) Find all possible cycle shapes of elements of S8 , and find the orders of the elements

with those cycle shapes.

(b) Find all possible cycle shapes of elements of A8 .

88

Groups

35. Show that the group of positive rational numbers under multiplica-

tion is not cyclic. Why does this prove that the group of nonzero

rationals under multiplication is not cyclic?

36. Consider the set {4, 8, 12, 16}. Show that this set is a group under

multiplication modulo 20 by constructing its Cayley table. What

is the identity element? Is the group cyclic? If so, find all of its

generators.

37. Give an example of a group that has exactly 6 subgroups (including

the trivial subgroup and the group itself). Generalize to exactly n

subgroups for any positive integer n.

38. Let m and n be elements of the group Z. Find a generator for the

group (m) n (n).

39. Suppose that a and b are group elements that commute. If lal is

finite and lbl infinite, prove that labl has infinite order.

40. Suppose that a and b belong to a group G, a and b commute, and lal

and 1bl are finite. What are the possibilities for labl?

41. Let u belong to a group and lat – 100 Find la981 and 1070

42. Let F and F’ be distinct reflections in D21. What are the possibilities

for IFF’|?

=

10. If a

43. Suppose that it is a subgroup of a group G and TH

belongs to G and aº belongs to H, what are the possibilities for lal?

44. Which of the following numbers could be the exact number of

elements of order 21 in a group: 21600, 21602, 21604?

45. If G is an infinite group, what can you say about the number of

elements of order 8 in the group? Generalize.

46. If G is a cyclic group of order n, prove that for every element a in G,

a” = e.

47. For each positive integer n, prove that C*, the group of nonzero

complex numbers under multiplication, has exactly Ø(n) elements

of order n.

48. Prove or disprove that H = {n E Zin is divisible by both 8 and 10

is a subgroup of Z. What happens if “divisible by both 8 and 10” is

changed to “divisible by 8 or 10?”

49. Suppose that G is a finite group with the property that every nons

is not trivial, prove that every nonidentity element of G has the

same order.

identity element has prime order (for example, D, and D3). If Z(G)

50. Prove that an infinite group must have an infinite number of

subgroups.

51. Let p be a prime. If a group has more than p – 1 elements of order p,

why can’t the group be cyclic?

4 Cyclic Groups

89

52. Suppose that G is a cyclic group and that 6 divides (Gl. How many

elements of order 6 does G have? If 8 divides [G], how many ele-

ments of order 8 does G have? If a is one element of order 8, list the

other elements of order 8.

53. List all the elements of Z40 that have order 10. Let Ixl = 40. List all

the elements of (x) that have order 10.

54. Reformulate the corollary of Theorem 4.4 to include the case when

the group has infinite order.

55. Determine the orders of the elements of D33 and how many there are

of each.

56. When checking to see if (2) = U(25) explain why it is sufficient

to check that 210 + 1 and 24 + 1.

57. If G is an Abelian group and contains cyclic subgroups of orders 4

and 5, what other sizes of cyclic subgroups must G contain?

Generalize.

58. If G is an Abelian group and contains cyclic subgroups of orders 4

and 6, what other sizes of cyclic subgroups must G contain?

Generalize.

59. Prove that no group can have exactly two elements of order 2.

60. Given the fact that U(49) is cyclic and has 42 elements, deduce the

number of generators that U(49) has without actually finding any of

the generators.

61. Let a and b be elements of a group. If lal = 10 and 1bl = 21, show

that (a) n (b) = {e}.

62. Let a and b belong to a group. If lal and \bl are relatively prime,

show that (a) n (b) = {e}.

63. Let a and b belong to a group. If lal = 24 and 1b 10, what are the

possibilities for l(a) n (b)?

64. Prove that U(2″) (n = 3) is not cyclic.

65. Prove that for any prime p and positive integer n, ” (p”) = p” – p-1.

66. Prove that Z has an even number of generators if n > 2. What does

this tell you about $(n)?

67. If las1 = 12, what are the possibilities for lal? If la41 – 12, what are

the possibilities for lal?

68. Suppose that Ixl = n. Find a necessary and sufficient condition on r

and s such that (x”) < (xs).
69. Let a be a group element such that |al = 48. For each part, find a
divisor k of 48 such that
a. (a21) = (ak);
b. (a14) = (ak);
c. (a18) = (a).