Math 4124
Monday, May 2
Solutions to Sample Final Problems
1. 3200
=
2
7
·
5
2
. The number of isomorphism classes of abelian groups of order 5
2
is
the number of partitions of 2. The partitions of 2 are 2 and (1,1), hence the number of
isomorphism classes of abelian groups of order 5
2
is 2. The number of isomorphism
classes of abelian groups of order 2
7
is the number of partitions of 7.
The parti
tions of 7 are 7, (6,1), (5,2), (4,3), (5,1,1), (4,2,1), (3,3,1), (3,2,2), (4,1,1,1), (3,2,1,1),
(2,2,2,1), (3,1,1,1,1), (2,2,1,1,1), (2,1,1,1,1,1), (1,1,1,1,1,1,1), which gives a total of
15 partitions. Therefore the number of isomorphism classes of abelian groups of order
3200 is 2
·
15
=
30. Since exactly one of these groups is cyclic, we conclude that the
number of noncyclic abelian groups is 29.
2. (a) This is not an ideal. For example 2 is in the described subset, but 2
x
is not.
(b) This is an ideal. Let
I
be described set. Then
I
is certainly an abelian group under
addition. Finally consider 4
a
0
+
2
a
1
x
+
a
2
x
2
+
···
+
a
n
x
n
, where
a
i
∈
Z
for all
i
;
this is the general element of
I
. The general element of
Z
[
x
]
is
b
0
+
b
1
x
+
···
+
b
m
x
m
, where
b
i
∈
Z
. When we multiply these two elements together, we get
4
a
0
b
0
+
2
(
a
1
b
0
+
2
a
0
b
1
)
x
+(
4
a
0
b
2
+
2
a
1
b
1
+
a
2
b
0
)
x
2
+
···
+
x
n
+
m
.
The above element is in
I
and it follows that
I
is an ideal of
Z
[
x
]
.
Finally let
4
c
0
+
2
c
1
x
+
c
2
x
2
+
···
+
c
k
x
k
be another element of
I
. Then
I
2
consists of sums
of elements of the form
16
a
0
c
0
+
8
(
a
1
c
0
+
a
0
c
1
)
x
+
4
(
a
2
c
0
+
a
1
c
1
+
a
0
c
2
)
x
2
+
2
(
2
a
3
c
0
+
a
2
c
1
+
a
1
c
2
+
2
a
0
c
3
)
x
3
+
···
+
a
n
c
k
x
n
+
k
.
It follows that
I
2
consists of all elements whose constant coefficient is a multiple
of 16, whose coefficient of
x
is a multiple of 8, whose coefficient of
x
2
is a multiple
of 4, and whose coefficient of
x
3
is a multiple of 2. The coefficients of
x
4
and
higher degree terms are arbitrary.
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 Spring '08
 Staff
 Math, Algebra, Ring theory, Group homomorphism, abelian groups, Ker θ

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