Notes: F.P. Greenleaf c
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Algebra I: Chapter 5. Permutation Groups
5.1 The Structure of a Permutation.
The
permutation group
S
n
is the collection of all bijective maps
σ
:
X
→
X
of the set
X
=
{
1
,
2
,...,n
}
, with composition of maps (
◦
) as the group operation. We introduced
permutation groups in Example 3.1.15 of Section 3, which you should review before
proceeding. There we introduced basic notation for describing permutations. The most
basic kind of permutation is a
cycle
. Recall that
5.1.1 Definition.
For
k>
1
, a
k
cycle
is a permutation
σ
= (
i
1
,...,i
k
)
that acts on
X
in the following way
(1)
σ
maps
braceleftbigg
i
1
→
i
2
→
...
→
i
k
→
i
1
(
a cyclic shift of list entries
)
j
→
j
for all
j
not in the list
{
i
1
,...,i
k
}
Onecycles
(
k
)
are redundant; every onecycle reduces to the identity map
id
X
, so we
seldom write them explicitly, though it is permissible and sometimes useful to do so.
The
support
of a
k
cycle is the set of entries
supp(
σ
) =
{
i
1
,...,i
k
}
, in no particular
order. The support of a 1cycle
(
k
)
is the onepoint set
{
k
}
.
Recall that the order of the entries in a cycle (
i
1
,...,i
k
) matters, but cycle notation is
somewhat ambiguous:
k
different symbols obtained by cyclic shifts of the entries in
σ
all
describe the same operation on
X
.
(
i
1
,...,i
k
) = (
i
2
,...,i
k
,i
1
) = (
i
3
,...,i
k
,i
1
,i
2
) =
...
= (
i
k
,i
1
,...,i
k
−
1
)
Thus (123) = (231) = (312) all specify the same operation 1
→
2
→
3
→
1 in
X
, and
likewise (
i,j
) = (
j,i
) for any 2cycle. If we mess up the cyclic order of the entries we
do not
get the same element in
S
n
, for example (123)
negationslash
= (132) as maps because the first
operation sends 1
→
2 while the second sends 1
→
3.
In Section 3.1 we also showed how to evaluate products
στ
of cycles, and noted the
following important fact.
(2)
If
σ
= (
m
1
,...,m
k
)
and
τ
= (
n
1
,...,n
r
)
are disjoint cycles, so that
supp(
σ
)
∩
supp(
τ
) =
{
m
1
,...,m
k
} ∩ {
n
1
,...,n
r
}
=
∅
then these operations commute
στ
=
τσ
. If supports overlap, the cycles may
or may not commute.
Since any 1cycle (
k
) is the identity operator, certain cycles with overlapping supports
such as (4) and (345) do commute, so property (2) only works in one direction; on
the other hand an easy calculation of the sort outlined in Example 3.1.15 shows that
(23)(345) = (2345), which is not equal to (345)(23) = (2453).
Our first task is to make good on a claim stated in 3.1.15: every permutation can
be written uniquely as a product of
disjoint commuting cycles
. This is a great help in
understanding how arbitrary permutations work.
5.1.2 Theorem.
Every
σ
∈
S
n
has a factorization
σ
=
producttext
r
i
=1
σ
i
into cycles whose
supports are disjoint and fill
X
(3)
X
=
r
uniondisplay
i
=1
supp(
σ
i
)
and
supp(
σ
i
)
∩
supp(
σ
j
)
for
i
negationslash
=
j
1
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Some factors may be trivial
1
cycles, which must be written down to get the support
condition
(3)
. The factors
σ
i
are uniquely determined, and they commute.
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 Spring '11
 ALgebra
 Algebra, Group Theory, Symmetric group, Sn, cycle type

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