This preview shows page 1. Sign up to view the full content.
Homework #3
Due: September 14, 2011
1. Show that the multiplicative group
Z
/
14
is cyclic. Find all of its generators.
2. In each case determine whether
G
is a cyclic group.
(a)
G
=
Z
/
7
(b)
G
=
Z
/
12
(c)
G
=
Z
/
16
(d)
G
=
Z
/
11
3. Find all of the generators of:
(a)
Z
5
(b)
Z
10
(c)
Z
16
(d)
Z
18
4. Let
G
be a group and let
a
2
G
be an element of order 20. Compute:
(a)
o
(
a
2
)
(b)
o
(
a
8
)
(c)
o
(
a
15
)
(d)
o
(
a
3
)
5. Let
a
be an element in a group
G
such that
a
15
=
e
. What are the possibilities for the
order of
a
?
6. Let
G
be a cyclic group of order 30 with generator
a
.
(a) List the elements of order 2 in
G
.
(b) List the elements of order 3 in
This is the end of the preview. Sign up
to
access the rest of the document.
Unformatted text preview: G . (c) List the elements of order 10 in G . 7. Determine (with explanation) if each of the following groups is cyclic: (a) Z 3 £ Z 9 (b) Z 3 £ Z 10 8. How many elements of order 4 does the group Z 4 £ Z 4 have? Explain why Z 4 £ Z 4 has the same number of elements of order 4 as does the group Z 400 £ Z 800 . Generalize to the case Z 4 m £ Z 4 n where m and n are arbitrary positive integers. Math 4023 1...
View
Full
Document
This document was uploaded on 12/28/2011.
 Fall '09

Click to edit the document details