3 Permutations d`un ensemble ni, groupe symétrique. Applications

E IdEE.
card (E)E.
S(E)S(E)E
S(E)
S(E)E.
ES(E)
{IdE}.
E={1,2,··· , n} ⊂ N,SnS(E)
n
σ∈ Sn,
σ=1 2 ··· n
σ(1) σ(2) ··· σ(n)
σ σ :k∈E7→ σ(k).
Sn.
12345
2145312345 35412=12345 43521
12345
21453−1
=12345
21534

σ∈ S (E)r, σrE
σr=
IdEr= 0
σ◦ ··· ◦ σ r r ≥1
(σ−r)−1r≤ −1
r2 card (E). r
r σ ∈ S (E)r E
{x1,··· , xr}E
∀k∈ {1,··· , r −1}, σ (xk) = xk+1
σ(xr) = x1
∀x∈E\ {x1,··· , xr}, σ (x) = x
σ= (x1,··· , xr)
{x1,··· , xr}σSupp (σ).
r
(x1, x2,···, xr),(x2, x3,··· , xr, x1), , (xr, x1,··· , xr−1)
r
r r
(x1, x2,··· , xr)−1= (xr, xr−1,··· , x1)
x0=xr,
σ(xk−1) = xk(1 ≤k≤r)
σ(x) = x x ∈E\ {x1,··· , xr}⇔σ−1(xk) = xk−1(1 ≤k≤r) σ−1(x) = x x ∈E\ {x1,··· , xr}
σ= (x1,··· , xr)r k
1r
xk=σk−1(x1)
k= 1 1 ≤k−1≤r−1,
xk=σ(xk−1) = σσk−2(x1)=σk−1(x1)
2.
τ2S(E),
τ̸=IdEτ2=IdE. τ−1=τ.
r r (S(E),◦).

σ= (x1,··· , xr)r r ≥2.
k1r,
σr(xk) = σrσk−1(x1)=σk−1(σr(x1))
=σk−1σσr−1(x1)=σk−1(σ(xr))
=σk−1(x1) = xk
σ(x) = x, x ∈E\ {x1,··· , xr}, σr=IdE.
σk−1(x1) = xk̸=x1,2≤k≤r, σk−1̸=IdEσ
r.
r σ r σ−1=σr−1.
σ r σmm
m r m =qr +i0≤i≤r−1
σm=σi.
E3S(E)
x1, x2, x3E τ1= (x1, x2), τ2= (x2, x3). τ2◦
τ1(x1) = x3τ1◦τ2(x1) = x2̸=x3. τ2◦τ1̸=τ1◦τ2S(E)
card (E) = n≥2.2≤r≤n, S(E)
Cr
n(r−1)! = n!
r(n−r)! r
r(x1,···, xr)E,
Ar
n=r!Cr n=n!
(n−r)! {x1,··· , xr}E, r
(x1,··· , xr),(x2, x3,··· , xr, x1), , (xr, x1,··· , xr−1)
Ar
n
r=
(r−1)!Cr
n
σ σ′Supp (σ)∩Supp (σ)
σσ′
σ= (x1, x2,··· , xr)σ′= (x′
1, x′ 2,··· , x′ s) Supp (σ)∩
Supp (σ) = {xk}. j 1s xk=x′
j,
σσ′= (xk+1,··· , xr, x1,··· , xk)xk, x′
j+1,··· , x′ s, x′ 1,··· , x′ j−1
=xk+1,··· , xr, x1,··· , xk, x′
j+1,··· , x′ s, x′ 1,···, x′ j−1
S(E)R∗
S(E)R∗.

r2 card (E).
S(E)r r r
σ= (x1, x2,···, xr)τ,
τ◦σ◦τ−1= (τ(x1), τ (x2),··· , τ (xr))
S(E),
σ σ′r τ
σ′=τ◦σ◦τ−1.
σ′′ = (τ(x1), τ (x2),··· , τ (xr)) , τ ◦σ=
σ′′ ◦τ.
x∈E\{x1,··· , xr}, σ (x) = x τ (x)∈E\{τ(x1),··· , τ (xr)},
τ◦σ(x) = τ(x) = σ′′ (τ(x)) = σ′′ ◦τ(x)
x xk,
τ◦σ(x) = τ(σ(xk)) = τ(xk+1)
xr+1 =x1
σ′′ ◦τ(x) = σ′′ (τ(xk)) = τ(xk+1)
τ◦σ=σ′′ ◦τ, τ ◦σ◦τ−1=σ′′.
σ= (x1, x2,··· , xr)σ′= (x′
1, x′ 2,··· , x′ r)r
φ E \ {x1,··· , xr}E\ {x′
1,··· , x′ r}, τ E
τ(xk) = x′
kk= 1,··· , r τ (x) = φ(x)x∈E\ {x1,···, xr}
τ◦σ◦τ−1= (τ(x1), τ (x2),···, τ (xr)) = (x′
1, x′ 2,··· , x′ r) = σ′
r2 card (E),
S(E)r
S(E)r
rAr
n
r= (r−1)!Cr
n.
Z(S(E)) S(E),
S(E)S(E).
Z(S(E)) = S(E) card (E) = 2
{IdE}card (E)≥3
card (E) = 2,S(E)Z(S(E)) = S(E).
card (E)≥3σ Z (S(E)) . x ̸=y E,
(σ(x), σ (y)) = σ(x, y)σ−1= (x, y)σσ−1= (x, y)
σ{x, y}={x, y}.card (E)≥3, x ∈E
y̸=z x {x}={x, y}∩{x, z},
{σ(x)}=σ({x}) = σ({x, y} ∩ {x, z})
=σ({x, y})∩σ({x, z}) = {x, y} ∩ {x, z}={x}
σ(x) = x. σ =IdE.
S(E){Id}.
S(E)n≥3.

Sn
card (E)≥3. σ ∈ S (E)\
{IdE}, σ. σ /∈Z(S(E))
Z(S(E)) = {IdE}.
σ∈ S (E)\ {Id}, x ∈E y =σ(x)̸=x.
z∈E\ {x, y}E3τ τ = (y, z).
στ (x) = σ(x) = y τσ (x) = τ(y) = z̸=y
στ ̸=τσ σ /∈Z(S(E)) .
Sn
E, F φ E F,
S(E)S(F)
ψ:S(E)→ S (F)
σ7→ φ◦σ◦φ−1
σ∈ S (E), ψ (σ)∈ S (F)σ1, σ2
S(E),
ψ(σ1◦σ2) = φ◦σ1◦σ2◦φ−1=φ◦σ1◦φ−1◦φ◦σ2◦φ−1
=ψ(σ1)◦ψ(σ2)
ψS(E)S(F).
σ∈ker (ψ), φ ◦σ◦φ−1=IdFσ=φ−1◦IdF◦φ=IdE, ψ
σ′∈ S (F), σ =φ−1◦σ′◦φS(E)ψ(σ) = σ′, ψ
E n
Sn{1,2,··· , n}.
n≥1E n, card (S(E)) = n!
E={x1,··· , xn}
σ∈ S (E), n σ (x1), σ (x1)n−1
σ(x2),··· ,1σ(xn), n!σ.
n≥1.
n= 1 S(E) = {IdE}.
n−1≥1E={x1,··· , xn}
n≥2
HS(E)E
xn. H S(E).
Id ∈H σ1, σ2H, σ1σ−1
2(xn) = σ1(xn) = xn, σ1σ−1 2∈H
HS(E).
σ∈H F ={x1,··· , xn−1}
HS(F).