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Math 4124
Wednesday, February 2
February 2, Ungraded Homework
Exercise 2.3.1 on page 60
Find all subgroups of
Z
45
=
h
x
i
, giving a generator for each.
Describe the containments between these subgroups.
The problem is equivalent to ﬁnding all the subgroups and containments between them for
Z
/
45
Z
. For each positive integer
n
dividing 45, there is a unique subgroup of order
n
;
this subgroup is
h
45
/
n
i
, the cyclic group with generator
45
/
n
. Therefore the subgroups of
Z
/
45
Z
are
h
1
i
,
h
3
i
,
h
5
i
,
h
9
i
,
h
15
i
,
h
45
i
.
For the containments, we have
h
a
i ⊆ h
b
i
if and only if
b
±
±
a
(this only works when
a
,
b
±
±
45).
Thus we have
h
45
i
is contained in
h
1
i
,
h
3
i
,
h
5
i
,
h
9
i
,
h
15
i
,
h
45
i
.
h
15
i
is contained in
h
1
i
,
h
3
i
,
h
5
i
,
h
15
i
.
h
9
i
is contained in
h
1
i
,
h
3
i
,
h
9
i
.
h
5
i
is contained in
h
1
i
,
h
5
i
.
h
3
i
is contained in
h
1
i
,
h
3
i
.
h
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This note was uploaded on 01/02/2012 for the course MATH 4124 taught by Professor Staff during the Spring '08 term at Virginia Tech.
 Spring '08
 Staff
 Math, Algebra

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