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58
(ii)
Find two nonisomorphic groups of order 6.
Solution.
A cyclic group of order 6 is not congruent to
S
3
, for the
former is abelian and
S
3
is not.
2.71
Prove that a dihedral group of order 4 is isomorphic to
V
, the 4group, and
a dihedral group of order 6 is isomorphic to
S
3
.
Solution.
Absent.
2.72
Prove that any two dihedral groups of order 2
n
are isomorphic.
Solution.
Absent.
2.73
De
f
ne a function
f
:
S
n
→
GL
(
n
,
R
)
by
f
:
σ
7→
P
σ
, where
P
σ
is the
matrix obtained from the
n
×
n
identity matrix
I
by permuting its columns
by
σ
(the matrix
P
σ
is called a
permutation matrix
). Prove that
f
is an
isomorphism from
S
n
to a subgroup of GL
(
n
,
R
)
.
Solution.
Absent.
2.74
(i)
Find a subgroup
H
≤
S
4
with
H
∼
=
V
but with
H
±=
V
.
Solution.
An example is
H
={
1
,(
12
), (
34
), (
1234
)
}
.
(ii)
Prove that the subgroup
H
in part (i) is not a normal subgroup.
Solution.
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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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