Question

58

204
Groups
69. In
D
55. In D, let K
Ro, D, D', R80. Show that K
{Ro, D} and let L
4
L<A D, but that K is not normal in D4. (Normality is not transitive
Compare Exercise 4, Supplementary Exercises for Chapters 5-8
con
56. Show that the intersection of two normal subgroups of G is a nor-
70. Pro
71. If
72. If
tair
mal subgroup of G. Generalize.
57. Give an example of subgroups H and K of a group G such that HK
is not a subgroup of G.
58. If N and M are normal subgroups of G, prove that NM is also a
normal subgroup of G.
59. Let N be a normal subgroup of a group G. If N is cyclic, prove that
every subgroup of N is also normal in G. (This exercise is referred
to in Chapter 24.)
60. Without looking at inner automorphisms of D, determine the num-
ber of such automorphisms.
61. Let H be a normal subgroup of a finite group G and let x E G. If
ged(lxl, IGIHI)
in Chapter 25.)
62. Let G be a group and let G' be the subgroup of G generated by the
73. Pr
74. Le
inc
an
75. Le
a s
ele
76. Su
1, show that x E H. (This exercise is referred to
or
77. Le
Sh
(xly xy I x, y E G. (See Exercise 3, Supplementary
set S
78. A
Exercises for Chapters 5-8, for a more complete description of G'.)
SU
a. Prove that G' is normal in G.
prг
b. Prove that G/G' is Abelian.
m
c. If GIN is Abelian, prove that G'<N.
d. Prove that if H is a subgroup of G and G' <H, then H is normal
79. Le
or
in G.
63. If N is a normal subgroup of G and IGINI
m, show that xm E N
1
for all x in G.
Suggested F
64. Suppose that a group G has a subgroup of order n. Prove that the
intersection of all subgroups of G of order n is a normal subgroup
Micha
nitely
(2000
of G.
65. If G is non-Abelian, show that Aut(G) is not cyclic.
The
66. Let IGI p"m, where p is prime and gcd(p, m) = 1. Suppose that
H is a normal subgroup of G of order p". If K is a subgroup of G of
order p, show that K C H.
67. Suppose that H is a normal subgroup of a finite group G. If GlH
has an element of order n, show that G has an element of order n.
Show, by example, that the assumption that G is finite is necessary
68. Recall that a subgroup N of a group G is called characteristic if
(N) = N for all automorphisms of G. If is a characteristic
subgroup of G, show that N is a normal subgroup of G.
pro
tex
J. A.C
Zn" F
Th
rel:
expand button
Transcribed Image Text:204 Groups 69. In D 55. In D, let K Ro, D, D', R80. Show that K {Ro, D} and let L 4 L<A D, but that K is not normal in D4. (Normality is not transitive Compare Exercise 4, Supplementary Exercises for Chapters 5-8 con 56. Show that the intersection of two normal subgroups of G is a nor- 70. Pro 71. If 72. If tair mal subgroup of G. Generalize. 57. Give an example of subgroups H and K of a group G such that HK is not a subgroup of G. 58. If N and M are normal subgroups of G, prove that NM is also a normal subgroup of G. 59. Let N be a normal subgroup of a group G. If N is cyclic, prove that every subgroup of N is also normal in G. (This exercise is referred to in Chapter 24.) 60. Without looking at inner automorphisms of D, determine the num- ber of such automorphisms. 61. Let H be a normal subgroup of a finite group G and let x E G. If ged(lxl, IGIHI) in Chapter 25.) 62. Let G be a group and let G' be the subgroup of G generated by the 73. Pr 74. Le inc an 75. Le a s ele 76. Su 1, show that x E H. (This exercise is referred to or 77. Le Sh (xly xy I x, y E G. (See Exercise 3, Supplementary set S 78. A Exercises for Chapters 5-8, for a more complete description of G'.) SU a. Prove that G' is normal in G. prг b. Prove that G/G' is Abelian. m c. If GIN is Abelian, prove that G'<N. d. Prove that if H is a subgroup of G and G' <H, then H is normal 79. Le or in G. 63. If N is a normal subgroup of G and IGINI m, show that xm E N 1 for all x in G. Suggested F 64. Suppose that a group G has a subgroup of order n. Prove that the intersection of all subgroups of G of order n is a normal subgroup Micha nitely (2000 of G. 65. If G is non-Abelian, show that Aut(G) is not cyclic. The 66. Let IGI p"m, where p is prime and gcd(p, m) = 1. Suppose that H is a normal subgroup of G of order p". If K is a subgroup of G of order p, show that K C H. 67. Suppose that H is a normal subgroup of a finite group G. If GlH has an element of order n, show that G has an element of order n. Show, by example, that the assumption that G is finite is necessary 68. Recall that a subgroup N of a group G is called characteristic if (N) = N for all automorphisms of G. If is a characteristic subgroup of G, show that N is a normal subgroup of G. pro tex J. A.C Zn" F Th rel:
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