Homework 4 Solutions
Math 332, Spring 2010
Problem 1.
Proposition.
If
n
≥
3
, then every element of
A
n
can be written as a product of one or more
3
cycles.
Proof.
Let
α
∈
A
n
. Since
α
is an even permutation, it is possible to express
α
as the product
of an even number of 2cycles
σ
1
,τ
1
,...,σ
k
,τ
k
, which we can group into pairs as follows:
α
=
σ
1
τ
1
σ
2
τ
2
···
σ
k
τ
k
= (
σ
1
τ
1
)(
σ
2
τ
2
)
···
(
σ
k
τ
k
)
.
We claim that each product
σ
i
τ
i
of two 2cycles can be expressed as the product of one or
more threecycles.
There are three cases. If the 2cycles
σ
i
and
τ
i
are disjoint, say
σ
i
= (
a b
) and
τ
i
= (
c d
),
then
σ
i
τ
i
= (
a b
)(
c d
) = (
a c d
)(
a b d
)
.
If the 2cycles
σ
i
and
τ
i
are equal, then
σ
i
τ
i
=
σ
i
2
=
e
= (1 2 3)(1 3 2)
.
The last possibility is that
σ
i
and
τ
i
have one number in common, say
σ
i
= (
a b
) and
τ
i
= (
a c
). In this case, the product is a single 3cycle:
σ
i
τ
i
= (
a b
)(
a c
) = (
a c b
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 Spring '09
 Math, Algebra, following table, Symmetric group, Conjugacy class, Conjugacy Class Identity, perfect riﬄe shuﬄe, σk τk

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