Quiz 3 Practice Problem Solutions
Math 332, Spring 2010
Questions on Groups
1.
The group
G/N
has four elements:
N
=
{
1
,
4
}
,
2
N
=
{
2
,
8
}
,
7
N
=
{
7
,
13
}
,
11
N
=
{
11
,
14
}
.
Since (2
N
)
2
= (7
N
)
2
= (11
N
)
2
=
N
, the isomorphism type is
Z
2
×
Z
2
(or
V
).
2.
Since
(
(1 2 3 4)
N
)
2
= (1 3)(2 4)
N
=
N
, the given element has
order 2
.
3.
Since
s
∈
N
and
r

1
sr
=
r
6
s
3
N
, the subgroup
N
is
not normal
.
4.
If
aN,bN
∈
G/N
, then (
aN
)(
bN
) = (
ab
)
N
= (
ba
)
N
= (
bN
)(
aN
), which proves that
G/N
is abelian.
5.
{
(0
,
0)
,
(1
,
1)
,
(2
,
2)
}
.
6.
If (
g
1
,h
1
)
,
(
g
2
,h
2
)
∈
G
×
H
, then
π
(
(
g
1
,h
1
)(
g
2
,h
2
)
)
=
π
(
g
1
g
2
,h
1
h
2
) =
g
1
g
2
=
π
(
g
1
,h
1
)
π
(
g
2
,h
2
,
which proves that
π
is a homomorphism.
7.
Clearly
S
is a subset of
G
, and
S
is nonempty since
e
∈
S
. Moreover, if
g
1
,g
2
∈
S
, then
ϕ
(
g
1
) =
g
1
and
ϕ
(
g
2
) =
g
2
, so
ϕ
(
g
1
g

1
2
) =
ϕ
(
g
1
)
ϕ
(
g
2
)

1
=
g
1
g

1
2
. It follows that
g
1
g

1
2