Quiz 1 Practice Problems: Cyclic Groups
Math 332, Spring 2010
These are not to be handed in. The quiz will be on Tuesday.
1.
Find all generators of
Z
6
,
Z
8
, and
Z
20
.
2.
List all elements of the subgroup
h
30
i
in
Z
80
.
3.
If

a

= 60, what is the order of
a
24
?
4.
How many subgroups does
Z
20
have? List the possible generators for each subgroup.
5.
Suppose that

a

= 24. Find a generator for
h
a
21
i ∩ h
a
10
i
. In general, what is a generator
for the subgroup
h
a
m
i ∩ h
a
n
i
?
6.
In
Z
60
, list all generators for the subgroup of order 12.
7.
Let
G
be a group and let
a
be an element of
G
.
a. If
a
12
=
e
, what can we say about the order of
a
?
b. If
a
m
=
e
, what can we say about the order of
a
?
8.
List all elements of order 8 in
Z
8000000
. How do you know your list is complete?
9.
Determine the subgroup lattice for
Z
p
2
q
, where
p
and
q
are distinct primes.
10.
Determine the subgroup lattice for
Z
p
n
, where
p
is a prime and
n
is some positive integer.
11.
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 Spring '09
 Math, Algebra, Prime number, Subgroup, Cyclic group

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