Math 4124
Wednesday, February 2
February 2, Ungraded Homework
Exercise 2.3.1 on page 60
Find all subgroups of
Z
45
=
x
, 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
45
/
n
, the cyclic group with generator
45
/
n
. Therefore the subgroups of
Z
/
45
Z
are
1 ,
3 ,
5 ,
9 ,
15 ,
45 .
For the containments, we have
a
⊆
b
if and only if
b a
(this only works when
a
,
b
45).
Thus we have
45
is contained in
1 ,
3 ,
5 ,
9 ,
15 ,
45 .
15
is contained in
1 ,
3 ,
5 ,
15 .
9
is contained in
1 ,
3 ,
9 .
5
is contained in
1 ,
5 .
3
is contained in
1 ,
3 .
1
is contained in
1 .
For the cyclic subgroup of order 45 with generator
x
, replace
n
with
x
n
everywhere. Thus,
for example, the subgroups of
x
are
x
,
x
3
,
x
5
,
x
9
,
x
15
,
x
45
. The last one is of
course 1 (the subgroup consisting of just the identity).
Exercise 2.3.3 on page 60
Find all generators for
Z
/
48
Z
.
Z
/
48
Z
=
{
0
,
1
,...,
47
}
.
Also
a
is a generator of
Z
/
48
Z
if and only if

a

=
48, and

a

=
48
/
(
48
,
a
)
.
Thus
a
is a generator of
Z
/
48
Z
if and only if
(
a
,
48
) =
1.
Therefore
the generators of
Z
/
48
Z
are
1
5
7
11
13
17
19
23
25
29
31
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 Spring '08
 Staff
 Math, Algebra, Greatest common divisor, Subgroup, Cyclic group, subgroups

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