2.2. SUBGROUPS AND CYCLIC GROUPS
101
Z
12
@
@
@
@
@
h
OE3Ł
i
h
OE2Ł
i
@
@
@
@
@
@
@
@
@
@
h
OE6Ł
i
h
OE4Ł
i
@
@
@
@
@
f
0
g
Figure 2.2.2.
Lattice of subgroups of
Z
12
.
integers
s
such that
s
is congruent modulo
n
to an integer multiple of
b
,
or, equivalently, the smallest of positive integers
s
such that
OEsŁ
2 h
OEbŁ
i
.
By the proof of Proposition
2.2.24
,
h
OEdŁ
i D h
OEbŁ
i
. The order of
OEbŁ
is the
order of
h
OEbŁ
i
, which is
n=d
, by Proposition
2.2.24
(b). Part (c) is left as
an exercise.
n
Example 2.2.29.
Find all generators of
Z
12
. Find all
OEbŁ
2
Z
12
such that
h
OEbŁ
i D h
OE3Ł
i
, the unique subgroup of order 4. The generators of
Z
12
are
exactly those
OEaŁ
such that
1
a
11
and
a
is relatively prime to 12.
Thus the generators are
OE1Ł; OE5Ł; OE7Ł; OE11Ł
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 Fall '08
 EVERAGE
 Algebra, Integers, Abelian group, Cyclic group, Cyclic Groups, hak, arbitrary cyclic groups

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