alex turner
13. Symmetric groups
13.1
13.2
13.3
Cycles, disjoint cycle decompositions
Adjacent transpositions
Worked examples
1. Cycles, disjoint cycle decompositions
The symmetric group Sn is the group of bijections of {1, . . . , n} to itself, also called permutations of n things. A standard notation for the permutation that sends i −→ i is
1
2
3
1
2
3
...
...
n n Under composition of mappings, the permutations of {1, . . . , n} is a group.
The fixed points of a permutation f are the elements i ∈ {1, 2, . . . , n} such that f (i) = i.
A k-cycle is a permutation of the form f ( 1) = for distinct this cycle:
1, . . . , k
2
f ( 2) =
...
3
f(
k−1 )
=
k
and f ( k ) =
among {1, . . . , n}, and f (i) = i for i not among the
(
1
2
3
...
j.
1
There is standard notation for
k)
Note that the same cycle can be written several ways, by cyclically permuting the can be written as
( 2 3 . . . k 1 ) or ( 3 4 . . . k 1 2 )
j:
for example, it also
Two cycles are disjoint when the respective sets of indices properly moved are disjoint. That is, cycles
( 1 2 3 . . . k ) and ( 1 2 3 . . . k ) are disjoint when the sets { 1 , 2 , . . . , k } and { 1 , 2 , . . . , k } are disjoint. [1.0.1] Theorem: Every permutation is uniquely expressible as a product of disjoint cycles.
191
192
Symmetric groups
Proof: Given g ∈ Sn , the cyclic subgroup g ⊂ Sn generated by g acts on the set X = {1, . . . , n} and decomposes X into disjoint orbits
Ox = {g i x : i ∈ Z}
for choices of orbit representatives x ∈ X. For each orbit representative x, let Nx be the order of g when restricted to the orbit g · x, and define a cycle
Cx = (x gx g 2 x . . . g Nx −1 x)
Since distinct orbits are disjoint, these cycles are disjoint. And, given y ∈ X, choose an orbit representative x such that y ∈ g · x. Then g · y = Cx · y. This proves that g is the product of the cycles Cx over orbit
13.1
13.2
13.3
Cycles, disjoint cycle decompositions
Adjacent transpositions
Worked examples
1. Cycles, disjoint cycle decompositions
The symmetric group Sn is the group of bijections of {1, . . . , n} to itself, also called permutations of n things. A standard notation for the permutation that sends i −→ i is
1
2
3
1
2
3
...
...
n n Under composition of mappings, the permutations of {1, . . . , n} is a group.
The fixed points of a permutation f are the elements i ∈ {1, 2, . . . , n} such that f (i) = i.
A k-cycle is a permutation of the form f ( 1) = for distinct this cycle:
1, . . . , k
2
f ( 2) =
...
3
f(
k−1 )
=
k
and f ( k ) =
among {1, . . . , n}, and f (i) = i for i not among the
(
1
2
3
...
j.
1
There is standard notation for
k)
Note that the same cycle can be written several ways, by cyclically permuting the can be written as
( 2 3 . . . k 1 ) or ( 3 4 . . . k 1 2 )
j:
for example, it also
Two cycles are disjoint when the respective sets of indices properly moved are disjoint. That is, cycles
( 1 2 3 . . . k ) and ( 1 2 3 . . . k ) are disjoint when the sets { 1 , 2 , . . . , k } and { 1 , 2 , . . . , k } are disjoint. [1.0.1] Theorem: Every permutation is uniquely expressible as a product of disjoint cycles.
191
192
Symmetric groups
Proof: Given g ∈ Sn , the cyclic subgroup g ⊂ Sn generated by g acts on the set X = {1, . . . , n} and decomposes X into disjoint orbits
Ox = {g i x : i ∈ Z}
for choices of orbit representatives x ∈ X. For each orbit representative x, let Nx be the order of g when restricted to the orbit g · x, and define a cycle
Cx = (x gx g 2 x . . . g Nx −1 x)
Since distinct orbits are disjoint, these cycles are disjoint. And, given y ∈ X, choose an orbit representative x such that y ∈ g · x. Then g · y = Cx · y. This proves that g is the product of the cycles Cx over orbit