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2.7. QUOTIENT GROUPS AND HOMOMORPHISM THEOREMS
143
Proof.
The statement is the special case of the Proposition with
G
D
G=K
and
'
W
G
!
G=K
the quotient map. Notice that applying the homomor
phism theorem again gives us the isomorphism
.G=N/=.K=N/
Š
G=K:
n
Example 2.7.16.
What are all the subgroups of
Z
n
? Since
Z
n
D
Z
=n
Z
,
the subgroups of
Z
n
correspond one to one with subgroups of
Z
containing
the kernel of the quotient map
'
W
Z
!
Z
=n
Z
, namely
n
Z
. But the
subgroups of
Z
are cyclic and of the form
k
Z
for some
k
2
Z
. So when
does
k
Z
contain
n
Z
? Precisely when
n
2
k
Z
, or when
k
divides
n
. Thus
the subgroups of
Z
n
correspond one to one with
positive integer divisors
of
n
. The image of
k
Z
in
Z
n
is cyclic with generator
ŒkŁ
and with order
n=k
.
Example 2.7.17.
When is there a surjective homomorphism from one
cyclic group
Z
k
to another cyclic group
Z
`
?
Suppose ﬁrst that
W
Z
k
!
Z
`
is a surjective homomorphism such
that
Œ1Ł
D
Œ1Ł
. Let
'
k
and
'
`
be the natural quotient maps of
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.
 Fall '08
 EVERAGE
 Algebra

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