98
2. BASIC THEORY OF GROUPS
2OE4Ł
D
OE8Ł
,
3OE4Ł
D
OE12Ł
,
4OE4Ł
D
OE2Ł
,
5OE4Ł
D
OE6Ł
,
6OE4Ł
D
OE10Ł
,
7OE4Ł
D
OE0Ł
. So
the order of
OE4Ł
is 7.
Example 2.2.19.
What is the order of
OE5Ł
in
˚.14/
? We have
OE5Ł
2
D
OE11Ł
,
OE5Ł
3
D
OE13Ł
,
OE5Ł
4
D
OE9Ł
,
OE5Ł
5
D
OE3Ł
,
OE5Ł
6
D
OE1Ł
, so the order of
OE5Ł
is 6.
Note that
j
˚.14/
j D
'.14/
D
6
, so this computation shows that
˚.14/
is
cyclic, with generator
OE5Ł
.
The next result says that cyclic groups are completely classified by
their order. That is, any two cyclic groups of the same order are isomor
phic.
Proposition 2.2.20.
Let
a
be an element of a group
G
.
(a)
If
a
has infinite order, then
h
a
i
is isomorphic to
Z
.
(b)
If
a
has finite order
n
, then
h
a
i
is isomorphic to the group
Z
n
.
Proof.
For part (a), define a map
'
W
Z
! h
a
i
by
'.k/
D
a
k
. This
map is surjective by definition of
h
a
i
, and it is injective because all powers
of
a
are distinct. Furthermore,
'.k
C
l/
D
a
k
C
l
D
a
k
a
l
. So
'
is an
isomorphism between
Z
and
h
a
i
.
For part (b), since
Z
n
has
n
elements
OE0Ł; OE1Ł; OE2Ł; : : : ; OEn
1Ł
and
h
a
i
has
n
elements
e; a; a
2
; : : : ; a
n
1
, we can define a bijection
'
W
Z
n
! h
a
i
by
'.OEkŁ/
D
a
k
for
0
k
n
1
. The product (addition) in
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 Fall '08
 EVERAGE
 Algebra, Cyclic group, Cyclic Groups, AK cl

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