Automorphismes de Sn

Sn
n∈N∗n6= 6 Sn
i.e. s 7→ σ◦s◦σ−1σ∈ Sn
•ϕ T
Snϕ
ϕ
Snσ∈ Sn
(a1, . . . , ak)k k
σ◦(a1, . . . , ak)◦σ−1= (σ(a1), . . . , σ(ak)) (∗)
k k
sSns1, . . . , sk
s(s=s1s2. . . sk)
σsσ−1= (σs1σ−1)(σs2σ−1). . . (σskσ−1) (∗∗)
i σsiσ−1si(∗){a1, . . . , ap}
si{σ(a1), . . . , σ(ap)}σsiσ−1σsiσ−1
ϕ
SnSnτ
τ2ϕ(τ) 2
ϕ(τ) 2
ppcm
TkSnk
2k≤n T1=T
(∗∗)TkSnϕ
ϕ(σ◦τ◦σ−1) = ϕ(σ)◦ϕ(τ)◦ϕ(σ)−1
ϕ(T1)Tk
n≥6ϕ(T1) = Tk
k6= 1 Tk|T1|=µn
2¶=n n−12
k≥2|Tk|=µn
2¶µn−2 2¶. . . µn−2k+ 2 2¶
k!
k k!
|Tk|=n(n−1) ...(n−2k+ 1)
2kk!
n≥6|Tk|=|T1|
k!2k= (n−2)(n−3) . . . (n−2k+ 1)
k1 2k≤n
k= 2 (n−2)(n−3) = 4 N
k≥3µn−k
k¶(n−2) . . . (n−k+ 1) = 2k−1n−k+ 1 = n−2
k= 3 (n−2)µn−3
3¶= 4
n= 6 ϕ(T1) = T1

•ϕ(T) = T(1, k)k∈ {2, . . . , n}
Sna1a2ϕ((1,2) = (a1, a2)
a b ϕ((1,3)) = (a, b) (1,2) (1,3) (a1, a2)
(a, b){a1, a2} ∩ {a, b} 6=∅a=a1
a3=b ϕ((1,3)) = (a1, a3)ϕ a26=a3
i∈ {2, . . . , n}ϕ((1, i)) (a1, ai)aiak
k < i i = 2 3 i > 3ϕ((1, i))
ϕ((1, k)) = (a1, ak) 2 ≤k≤i−1
(1, i) (1, k)i= 3
a1ϕ((1, i)) i2≥2i= 4
ϕ((1,4)) = (a2, a3)=(a1, a3)(a2, a1)(a1, a3) = ϕ((3,1)(1,2)(1,3))
(1,4) = (3,1)(1,2)(1,3)
a1ϕ((1, i)) (a1, ai)ϕ ai
akk < i
•n{1, . . . , n}a1, . . . , anϕ((1, i)) =
(a1, ai) 2 ≤i≤n σ i ∈ {1, . . . , n}ai
σ◦(1, i)◦σ−1= (a1, ai)
ϕ ϕσ:s7→ σ◦s◦σ−1
(1, i) 2 ≤i≤n ϕ (1, i)
Snϕ=ϕσ