Homework 6 Solutions
1. Table of Elements and their Orders:
element
1
7
43
49
51
57
93
99
101
107
143
149
151
157
193
199
order
1
4
4
2
2
4
4
2
2
4
4
2
2
4
4
2
Since

G

= 16 and
G
only has elements of 4 or 2 (or 1) we know either
G
∼
=
Z
4
×
Z
4
or
G
∼
=
Z
4
×
Z
2
×
Z
2
.
Z
4
×
Z
2
×
Z
2
has 8 elements of order 4 yet
Z
4
×
Z
4
has 12 elements of
order 4. Thus we can conclude
G
∼
=
Z
4
×
Z
2
×
Z
2
.
2. (a) If
G
is an abelian group of order 9 then
G
is isomorphic to either
Z
9
or
Z
3
×
Z
3
. The
former has 6 elements of order 9, 2 elements of order 3 and an identity. The latter has
8 elements of order 3 and an identity. Thus if you calculate the order of 3 elements
(besides the identity) you can deﬁnitively determine the isomorphism class of
G
.
(b) If
G
is an abelian group of order 18 then
G
is isomrphic to either
Z
2
×
Z
3
×
Z
3
or
Z
2
×
Z
9
.
The former has 8 elements of order 3, 1 element of order 2, 8 elements of order 6 and an
identity. The latter has 2 elements of order 3, 6 elements of order 9, 1 element of order
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 Spring '11
 Johnson
 Algebra, Cyclic group, Group isomorphism

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