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Quiz 3 Practice Problems
Math 332, Spring 2010
Questions on Groups
1.
Let
G
=
U
(15), and let
N
=
{
1
,
4
}
. Determine the isomorphism type of
G/N
.
2.
Let
G
=
S
4
, and let
N
=
{
e,
(1 2)(3 4)
,
(1 3)(2 4)
,
(1 4)(2 3)
}
. Determine the order of
the element (1 2 3 4)
N
in
G/N
.
3.
Let
G
=
D
8
, and let
N
=
{
e, s, r
4
, r
4
s
}
. Is
N
is a normal subgroup of
D
8
? Explain.
4.
Let
G
be an abelian group, and let
N
be a normal subgroup of
G
. Prove that
G/N
is
abelian.
5.
Let
ϕ
:
Z
3
×
Z
3
→
Z
3
be a homomorphism, and suppose that
ϕ
(1
,
0) = 1 and
ϕ
(0
,
1) = 2.
List the elements in the kernel of
ϕ
.
6.
Let
G
and
H
be groups, and deﬁne a function
π
:
G
×
H
→
G
by
π
(
g,h
) =
g.
Prove that
π
is a homomorphism.
7.
Let
G
be a group, let
ϕ
:
G
→
G
be a homomorphism, and let
S
=
{
g
∈
G

ϕ
(
g
) =
g
}
.
Prove that
S
≤
G
.
8.
List all isomorphism types of abelian groups of order 900.
9.
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 Spring '09
 Math, Algebra

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