Math 4124
Wednesday, April 6
April 6, Ungraded Homework
Exercise 5.2.1 on page 165. In each of parts (a) to (d), give the number of nonisomorphic
abelian groups of the speciﬁed order. (a) 100
(b) 576
(c) 1155
(d) 42875
(a) First write 100 as a product of prime factors; thus 100
=
2
2
5
2
. The number of partitions
of 2 is 2, namely
{
2
}
and
{
1,1
}
, thus the number of isomorphism classes of abelian
groups of order
p
2
is 2. It follows that the number of isomorphism classes of groups of
order 100 is 2
·
2
=
4.
(b) 576
=
2
6
3
2
. The partitions of 6 are
{
6
}
,
{
5,1
}
,
{
4,2
}
,
{
4,1,1
}
,
{
3,3
}
,
{
3,2,1
}
,
{
3,1,1,1
}
,
{
2,2,2
}
,
{
2,2,1,1
}
,
{
2,1,1,1,1
}
,
{
1,1,1,1,1,1
}
; thus there are 11 partitions of 6. Also
there are 2 partitions of 2. Therefore the number of isomorphism classes of abelian
groups of order 576 is 2
·
11
=
22.
(c) 1155
=
3
·
5
·
7
·
11. The number of partitions of 1 is 1. Therefore the number of isomor-
phism classes of abelian groups of order 1155 is 1
·
1
·
1
·
1
=
1.
(d) 42875