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46
(ii)
If
kr
≤
n
, where 1
<
r
≤
n
, prove that the number of permuta
tions
α
∈
S
n
, where
α
is a product of
k
disjoint
r
cycles, is
1
k
!
1
r
k
[
n
(
n
−
1
)
···
(
n
−
+
1
).
]
Solution.
Absent.
2.25
(i)
If
α
is an
r
cycle, show that
α
r
=
(
1
)
.
Solution.
If
α
=
(
i
0
...
i
r
−
1
)
, then the proof of Lemma 2.25(ii)
shows that
α
k
(
i
0
)
=
i
k
, where the subscript is read mod
r
. Hence,
α
r
(
i
0
)
=
i
0
. But the same is true if we choose the notation for
α
having any of the other
i
j
as the
f
rst entry.
(ii)
If
α
is an
r
cycle, show that
r
is the least positive integer
k
such
that
α
k
=
(
1
)
.
Solution.
Use Proposition 2.24. If
k
<
r
, then
α
k
(
i
0
)
=
i
k
±=
i
0
,
so that
α
k
±=
1.
2.26
Show that an
r
cycle is an even permutation if and only if
r
is odd.
Solution.
In the proof of Proposition 2.35, we showed that any
r
cycle
α
is a product of
r
−
1 transpositions. The result now follows from Proposi
tion 2.39, for sgn
(α)
=
(
−
1
)
r
−
1
=−
1.
2.27
Given
X
={
1
,
2
,...,
n
}
, let us call a permutation
τ
of
X
an
adjacency
if
it is a transposition of the form
(
ii
+
1
)
for
i
<
n
.I
f
i
<
j
, prove that
(
ij
)
is a product of an odd number of adjacencies.
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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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