THE UNIVERSITY OF SYDNEY
Math2968 Algebra (Advanced)
Semester 2
Tutorial Solutions Week 3
2008
1.
Write down the inverses of the following permutations as products of disjoint cycles:
α
= (1 2 3 4 5 6)
,
β
= (1 2)(3 4)(5 6 7 8)
,
γ
= (1 3 5)(2 4 6)
Express each of
α
,
β
,
γ
as a product of transpositions.
Solution
α

1
= (6 5 4 3 2 1)
,
β

1
= (1 2)(3 4)(8 7 6 5)
,
γ

1
= (5 3 1)(6 4 2)
α
= (1 2)(1 3)(1 4)(1 5)(1 6)
,
β
= (1 2)(3 4)(5 6)(5 7)(5 8)
,
γ
= (1 3)(1 5)(2 4)(2 6)
2.
Let
α
and
β
be permutations of a finite set. Verify that if (
x
1
x
2
. . . x
k
) is a cycle in the decomposition
of
α
then (
x
1
β x
2
β . . . x
k
β
) is a cycle in the decomposition of
β

1
αβ
.
Solution
Observe that, if (
x
1
x
2
. . . x
k
) is a cycle in the decomposition of
α
then, for each
i
,
(
x
i
β
)(
β

1
αβ
) = (
x
i
α
)
β
=
x
i
+1
β
(where we put
k
+ 1
≡
1), which shows (
x
1
β x
2
β . . . x
k
β
) is a cycle in the decomposition of
β

1
αβ
.
3.
Use the result of the previous exercise to write down
β

1
αβ
immediately when
(a)
α
= (1 2 3 4 5 6) ,
β
= (1 6)(2 5)(3 4) .
(b)
α
= (1 6)(2 5)(3 4) ,
β
= (1 2 3 4 5 6) .
(c)
α
= (1 9)(4 2 5 6)(8 3 7) ,
β
= (2 3 8 6 5 4)(9 1) .
(d)
α
= (2 3 8 6 5 4)(9 1) ,
β
= (1 9)(4 2 5 6)(8 3 7) .
Solution
(a)
β

1
αβ
= (6 5 4 3 2 1)
(b)
β

1
αβ
= (2 1)(3 6)(4 5)
(c)
β

1
αβ
= (1 9)(2 3 4 5)(6 8 7)
(d)
β

1
αβ
= (5 7 3 4 6 2)(1 9)
4.
Which of the following permutations are even? odd?
α
= (1 2)(3 4)
,
β
= (1 2 3 4 5)
,
γ
= (1 2 3 4 5 6)
δ
= (1 2 3 4 5)(1 3 5)(2 5 6 7)
,
ε
= (1 2)(1 3)(1 4 5)(4 2)
Solution
Here
α
and
β
are even, whilst
γ
,
δ
and
ε
are odd.
5.
Let
α
be a permutation of
{
1
,
2
, . . . , n
}
. Explain why
α

1
=
α
k
for some positive integer
k
. Explain
briefly why the set of all odd permutations of
{
1
,
2
, . . . , n
}
does not form a permutation group.
Solution
If
α
= 1 then
α

1
=
α
1
. Suppose
α
negationslash
= 1. Then
α
=
α
1
α
2
. . . α
ℓ
where
α
1
, . . . , α
ℓ
are disjoint cycles of
length
n
1
, . . . , n
ℓ
respectively, and
n
1
>
1. Then
α
n
1
...n
ℓ
= 1
,
giving
α

1
=
α
k
where
k
= (
n
1
. . . n
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 One '09
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 Algebra, Permutations, Vector Space, positive integer, nonzero real number

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