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60
2.81
De
f
ne
W
=h
(
12
)(
34
)
i
, the cyclic subgroup of
S
4
generated by
(
)(
)
.
Show that
W
is a normal subgroup of
V
, but that
W
is not a normal sub
group of
S
4
. Conclude that normality is not transitive:
K
C
H
and
H
C
G
need not imply
K
C
G
.
Solution.
Since every subgroup of an abelian group is a normal subgroup,
W
is a normal subgroup of
V
.However
,
W
is not a normal subgroup of
S
4
,
for conjugating
(
)(
)
by
(
13
)
gives
(
)(
)(
)(
)
=
(
23
)(
14
)/
∈
W
.
2.82
Let
G
be a
f
nite group written multiplicatively. Prove that if

G

is odd,
then every
x
∈
G
has a square root. Conclude, using Exercise 2.45, that
there exists exactly one
g
∈
G
with
g
2
=
x
.
Solution.
The function
ϕ
:
G
→
G
,de
f
ned by
ϕ(
g
)
=
g
2
, is a homo
morphism because
G
is abelian. Now ker
ϕ
={
g
∈
G
:
g
2
=
1
}={
1
}
because
G
has odd order, hence has no elements of order 2. It follows that
ϕ
is injective. By Exercise 2.13, the function
ϕ
is surjective; that is, for
each
x
∈
G
, there is
g
∈
G
with
x
=
g
)
=
g
2
. We have shown that
x
has a square root. This square root is unique, for if
g
2
=
x
=
h
2
for some
h
∈
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This note was uploaded on 12/21/2011 for the course MAS 4301 taught by Professor Keithcornell during the Fall '11 term at UNF.
 Fall '11
 KeithCornell

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