B U Department of Mathematics
Fall 2008 Math 321  Algebra,
First Midterm Exam, 14/11/2008, 17:0019:00
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over 100
In what follows,
Z
n
is the additive group with the modular addition;
G
,
H
and
K
are groups.
1. Solve the following questions within the space allocated. Do not write in the margins or wheresoever. Expain
your answer. These are not just yes/no questions. Figures are from Wikipedia.
(a) [10pts] Let
f
:
G
→
H
and
g
:
H
→
K
be isomorphisms. Is it true that
g
◦
f
is an isomorphism?
(b) [10pts] Give an example of two nonisomorphic groups of the same order. Explain.
(c) [10pts] Find
all
subgroups of
Z
15
. Explain why.
(d) [5pts] What is the order of the group of symmetries of a cube?
(e) [5pts] What is the order of the group of symmetries of a dodecahedron?
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2. We define an operation
·
on
G
×
H
=
{
(
g, h
)

g
∈
G, h
∈
H
}
as follows:
(
g
1
, h
1
)
·
(
g
2
, h
2
)
.
= (
g
1
g
2
, h
1
h
2
)
.
(a) [10pts] Show that
(
G
×
H,
·
)
is a group.
(b) [5pts] Let
g
∈
G
and
h
∈
H
have orders
m
and
n
respectively. What is the order of
(
g, h
)
in
G
×
H
?
(c) [5pts] Is
Z
2
×
Z
3
isomorphic to
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 Spring '11
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 Algebra, Addition, Normal subgroup, 10pts, arbitrary subgroup, U Department of Mathematics, A2 A3

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