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Math 4124
Monday, February 21
First Test. Answer All Problems.
Please Give Explanations For Your Answers
1. Prove that
Q
8
×
S
3
is
not
isomorphic to
D
48
. (
Q
8
denotes the quaternion group of
order 8 and
D
48
denotes the dihedral group of order 48.)
(12 points)
2. Let
G
be a group. Prove that the formula
(
g
,
h
)
·
x
=
gxh

1
for
g
,
h
,
x
∈
G
, deﬁnes an
action of
G
×
G
on
G
. Show further that if Z
(
G
)
±
=
1, then there exists a nonidentity
element which acts trivially on
G
(i.e. there exists 1
±
=
k
∈
G
×
G
such that
k
·
x
=
x
for
all
x
∈
G
).
(12 points)
3. Prove that if
A
and
B
are subsets of the group
G
with
A
⊆
B
, then C
G
(
B
)
±
C
G
(
A
)
. Is it
always true that N
G
(
B
)
±
N
G
(
A
)
? Justify your answer. (C
G
and N
G
denote centralizer
and normalizer)
(12 points)
4. Let
D
16
=
²
r
,
s

r
8
=
1
,
s
2
=
1
,
rs
=
sr

1
³
, the dihedral group of order 16. Determine
for which positive integers
n
there is a subgroup of order
n
in
D
16
(so you will need
to give an example of a subgroup of order
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 Spring '08
 Staff
 Math, Algebra

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