Math 4124
Wednesday, February 16
Fourth Homework Solutions
1. 2.3.6 on page 60. In
Z
/
48
Z
write out all elements of
h
a
i
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.
h
0
i
=
{
0
}
Order 1
h
1
i
=
{
0
,
1
,
2
, . . . ,
47
}
order 48
h
2
i
=
{
0
,
2
,
4
, . . . ,
46
}
order 24
h
3
i
=
{
0
,
3
,
6
, . . . ,
45
}
order 16
h
4
i
=
{
0
,
4
,
8
, . . . ,
44
}
order 12
h
6
i
=
{
0
,
6
,
10
, . . . ,
42
}
order 8
h
8
i
=
{
0
,
8
,
12
, . . . ,
40
}
order 6
h
12
i
=
{
0
,
12
,
24
,
36
}
order 4
h
16
i
=
{
0
,
16
,
32
}
order 3
h
24
i
=
{
0
,
24
}
order 2
Then we have
h
1
i
=
h
5
i
=
h
7
i
=
h
11
i
=
h
13
i
=
h
17
i
=
h
19
i
=
h
23
i
=
h
25
i
=
h
29
i
=
h
31
i
=
h
35
i
=
h
37
i
=
h
41
i
=
h
43
i
=
h
47
i
h
2
i
=
h
10
i
=
h
14
i
=
h
22
i
=
h
26
i
=
h
34
i
=
h
38
i
=
h
46
i
h
3
i
=
h
9
i
=
h
15
i
=
h
21
i
=
h
27
i
=
h
33
i
=
h
39
i
=
h
45
i
h
4
i
=
h
20
i
=
h
28
i
=
h
44
i
h
6
i
=
h
18
i
=
h