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5.5. TRANSITIVE SUBGROUPS OF
S
5
259
(a)
Show that ker
.±/
is generated by an element of
Z
2
±
Z
2
of order
2
.
(b)
Show that
±
is determined by its kernel. That is, if
±
and
±
0
are
two such homomorphisms with the same kernel, then
±
D
±
0
.
(c)
Show that if
ˇ
and
±
are two such homomorphisms, then there
exists an automorphism
'
2
Aut
.
Z
2
±
Z
2
/
such that ker
.ˇ/
D
ker
.±
ı
'/
. Consequently,
ˇ
D
±
ı
'
.
(d)
Conclude from Exercise
5.4.13
that if
ˇ
and
±
are two nontrivial
homomorphisms
from
Z
2
±
Z
2
into
Aut
.
Z
7
/
,
then
Z
7
Ì
ˇ
.
Z
2
±
Z
2
/
Š
Z
7
Ì
±
.
Z
2
±
Z
2
/:
5.4.15.
Classify the nonabelian group(s) of order 20.
5.4.16.
Classify the nonabelian group(s) of order 18.
5.4.17.
Classify the nonabelian group(s) of order 12.
5.4.18.
Let
p
be the largest prime dividing the order of a ﬁnite group
G
,
and let
P
be a
p
–Sylow subgroup of
G
. Find an example showing that
P
need not be normal in
G
.
5.5. Application: Transitive Subgroups of
S
5
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 Fall '08
 EVERAGE
 Algebra

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