abstract algerbra homework

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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).