Math 4124
Wednesday, February 16
Fourth Homework Solutions
1. 2.3.6 on page 60. In
Z
/
48
Z
write out all elements of
a
for every
a
. Find all inclu-
sions between subgroups in
Z
/
48
Z
.
Since 48 = 3 *16, 3 has two divisors and 16 has five divisors, we see that there are 2*5
= 10 subgroups.
0
=
{
0
}
Order 1
1
=
{
0
,
1
,
2
,...,
47
}
order 48
2
=
{
0
,
2
,
4
,...,
46
}
order 24
3
=
{
0
,
3
,
6
,...,
45
}
order 16
4
=
{
0
,
4
,
8
,...,
44
}
order 12
6
=
{
0
,
6
,
10
,...,
42
}
order 8
8
=
{
0
,
8
,
12
,...,
40
}
order 6
12
=
{
0
,
12
,
24
,
36
}
order 4
16
=
{
0
,
16
,
32
}
order 3
24
=
{
0
,
24
}
order 2
Then we have
1
=
5
=
7
=
11
=
13
=
17
=
19
=
23
=
25
=
29
=
31
=
35
=
37
=
41
=
43
=
47
2
=
10
=
14
=
22
=
26
=
34
=
38
=
46
3
=
9
=
15
=
21
=
27
=
33
=
39
=
45
4
=
20
=
28
=
44
6
=
18
=
30
=
42
8
=
40
12
=
36
16
=
32
0
,
24 are on their own.
The lattice of subgroups looks like (where we have labeled each subgroup with its
order)