Math 541
Solutions to HW #5
The following are from Gallian, Chapter 3 (6th edition).
In this solution set,

a

denotes the order of the element
a
(i.e. ord(
a
)).
•
# 1
:
–
Z
12
:
* 
Z
12

= 12;
* 
0

= 1;

1

= 12;

2

= 6;

3

= 4;

4

= 3;

5

= 12;

6

= 2;

7

= 12;

8

= 3;

9

= 4;

10

= 6;

11

= 12.
–
U
(10):
* 
U
(10)

=
φ
(10) =
φ
(2
·
5) = (1)(4) = 4;
* 
1

= 1;

3

= 4;

7

= 4;

9

= 2.
–
U
(12):
* 
U
(12)

=
φ
(12) =
φ
(2
2
·
3) = (2)(2) = 4;
* 
1

= 1;

5

= 2;

7

= 2;

11

= 2.
–
U
(20):
* 
U
(20)

=
φ
(20) =
φ
(2
2
·
5) = (2)(4) = 8;
* 
1

= 1;

3

= 4;

7

= 4;

9

= 2;

11

= 2;

13

= 4;

17

= 4;

19

= 2.
–
D
4
:
* 
D
4

= 8;
* 
R
0

= 1;

R
90

= 4;

R
180

= 2;

R
270

= 4;

H

= 2;

V

= 2;

D

= 2;

D’

= 2.
–
Observations
: The order of every element divides the order of the group.
•
# 2
:
–
In
Q
, we have
h
1
2
i
=
{
0
,
±
1
2
,
±
1
,
±
3
2
,
±
2
,
±
5
2
,...
}
–
In
Q
*
, we have
h
1
2
i
=
{
...
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 Fall '09
 Pollack
 Math, Algebra, Abelian, =, TwoStep Subgroup Test.

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