Math 3124
Tuesday, September 27
First Test. Answer All Problems.
Please Give Explanations For Your Answers.
1. Which of the following formulae for
x
*
y
defines an associative operation on
Z
#
4
, the
set
{
[
1
]
,
[
2
]
,
[
3
]
}
.
(a)
x
*
y
=
x
y
(b)
x
*
y
=
y
(10 points)
2. Let
α
=
(4 6 7)

1
(3 4)(1 2 3 6 5). Write
α
as a product of disjoint cycles. Also
determine whether
α
∈
A
7
.
(10 points)
3. Let
G
=
S
4
and
T
=
{
1
,
3
}
. Write all the elements of
G
T
and
G
(
T
)
, as a product of
disjoint cycles. (Recall that
G
T
=
{
α
∈
G

α
(
t
) =
t
for all
t
∈
T
}
and
G
(
T
)
=
{
α
∈
G

α
(
T
) =
T
}
.)
(10 points)
4. Let
G
be a group, let
X
⊆
G
, and let
C
=
{
g
∈
G

gx
=
xg
for all
x
∈
X
}
. Prove that
C
is a subgroup of
G
.
(10 points)
5. Let
G
consist of the three matrices
[
1
]
[
0
]
[
0
]
[
1
]
,
[
1
]
[
1
]
[
0
]
[
1
]
,
[
1
]
[
2
]
[
0
]
[
1
]
where the
entries are in
Z
3
. Compute the Cayley table of
G
with the operation matrix modular
multiplication.
(10 points)
6. Compute the order of
(
(
1 2 3 4
)(
5 6 7 8 9 10
)
,
[
1
]
)
in
S
10
×
(
Z
10
,
⊕
)
.
(10 points)
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Math 3124
Tuesday, September 27
First Test Solutions
1. (a) This is not an associative operation, in fact it is not even an operation. For example
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 Fall '08
 PARRY
 Math, Algebra, Addition, transpositions, Semigroup, associative operation

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