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45
(vii)
Every transposition is an even permutation.
Solution.
False.
(viii)
If a permutation
α
is a product of 3 transpositions, then it cannot
be a product of 4 transpositions.
Solution.
True.
(ix)
If a permutation
α
is a product of 3 transpositions, then it cannot
be a product of 5 transpositions.
Solution.
False.
(x)
If
σασ
−
1
=
ωαω
−
1
, then
σ
=
ω
.
Solution.
False.
2.22
Find sgn
(α)
and
α
−
1
, where
α
=
µ
123456789
987654321
¶
.
Solution.
In cycle notation,
α
=
(
19
)(
28
)(
37
)(
46
)
. Thus,
α
is even, being
the product of four transpositions. Moreover, being a product of disjoint
transpositions,
α
=
α
−
1
.
2.23
If
σ
∈
S
n
f
xes some
j
, where 1
≤
j
≤
n
(that is,
σ(
j
)
=
j
), de
f
ne
σ
0
∈
S
X
, where
X
={
1
,...,
b
j
n
}
,by
σ
0
(
i
)
=
i
)
for all
i
±=
j
.
Prove that
sgn
(σ
0
)
=
sgn
(σ).
Solution.
One of the cycles in the complete factorization of
σ
is the 1
cycle
(
j
)
. Hence, if there are
t
cycles in the complete factorization of
σ
,
then there are
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 Fall '11
 KeithCornell

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