Les groupes finis d`ordre 60 et 168

G60 = 223 5
npp G 1< n5= 5k+ 1|12
n5= 6 Ω 5 G G
P, Q ∈ΩP6=Q P, Q 'C5
u:P−→ Ω\ {P}
g−→ gQg−1
P6=Q gQg−1∈Ω\ {P}g∈P
u(g) = u(g0)g, g0∈P gg−1∈G(Q)Q
5G(Q)gg−16=e gg−1∈Q
P∩Q={e}g=g0u
α P P =< α >
ψ:1(F5) = F5∪ {∞} −→ Ω
ψ(∞) = P ψ(k) = αkQα−kk∈F5ψ(0) = Q
GΩ
ϕ:G−→ SΩ
ϕ(G)⊂AΩ'A6
g∈G≤6
ψ G ΩG1(F5)
ψ(g.x) = g.ψ(x).g−1
x∈1(F5)g∈G G A1(F5)
α α :x−→ x+ 1 1(F5)
ψ(α.∞) = α.P.α−1=P=ψ(∞)
k∈F5
ψ(α.k) = α(αkQα−k)α−1=αk+1Qα−k−1=ψ(k+ 1)
α1(F5)∞1(F5)\ {∞} =F5α
(0,1,2,3,4)
n5= 6 G(P)C10 D5
G≤6G(P)'D5M=G(P)∩G(Q)
M=< β >={, β}
G(P) = PoM=< α, β >

α β βαβ−1=α−1β β :x−→ −x
1(F5)
ψ(β.∞) = β.P.β−1=P=ψ(∞)
k∈F5
ψ(β.k) = β(αkQα−k)β−1= (βαkβ−1)(βQβ−1)(βα−kβ−1) = α−kQαk=ψ(−k)
Q=<−1>={1,−1} ⊂ F?
5F? 5N=F? 5\Q=
{2,3}β1(F5) 0 ∞Q N
β(1,4)(2,3)
βG(M) = G(β)G(β)SΩ(β)
G C4
C2×C2GAΩG
G(M) =< β, γ >
β γ βγ =γβ
G(P)G H G
G(P)H H G(P)G
G=< α, β, γ >
GΩ1(F5)G∞=G(P) =< α, β >
γ(∞)6=∞βγ(∞) = γβ(∞) = γ(∞)γ(∞)β
γ(∞)=0 γ∞
β1(F5) (0,∞)
γ a, b ∈1(F5)\ {0,∞} =F?
5k∈F? 5
βγ(k) = γβ(k) = γ(−k) = −γ(k)
γ(a) = a γ(−a) = −a{a, b}N={2,3}
Q={1,4}γ= (2,3)(0,∞)βγ = (1,4)(0,∞)
γ γβ γ = (1,4)(0,∞)γ
x−→ − 1
x1(F5)
G'2(F5)
G168 = 233 7
npp G 1< n7= 7k+ 1|24
n7= 8 Ω 7 G G
P, Q ∈ΩP6=Q P, Q 'C7
u:P−→ Ω\ {P}
g−→ gQg−1
P6=Q gQg−1∈Ω\ {P}g∈P
u(g) = u(g0)g, g0∈P gg−1∈G(Q)Q

7G(Q)gg−16=e gg−1∈Q
P∩Q={e}g=g0u
α P P =< α >
ψ:1(F7) = F7∪ {∞} −→ Ω
ψ(∞) = P ψ(k) = αkQα−kk∈F7ψ(0) = Q
GΩ
ϕ:G−→ SΩ
ϕ(G)⊂AΩ'A8
g∈G≤15
ψ G ΩG1(F7)
ψ(g.x) = g.ψ(x).g−1
x∈1(F7)g∈G G A1(F7)
α α :x−→ x+ 1 1(F7)
ψ(α.∞) = α.P.α−1=P=ψ(∞)
k∈F7
ψ(α.k) = α(αkQα−k)α−1=αk+1Qα−k−1=ψ(k+ 1)
α1(F7)∞1(F7)\ {∞} =F7α
(0,1,2,3,4,5,6)
n7= 8 G(P)G
≤15 M=G(P)∩G(Q)M
(G(P).G(Q)) = 212>168 M
M=< β >
G(P) = PoM=< α, β >
α β βαβ−1=αrr6≡ 0,1 7 βαkβ−1=αrk
β3= 1 α=αr3r3≡1 7 r= 2 r= 4
r= 4 βαβ−1=α4β2αβ−2=α2β M
β2
βαβ−1=α2
β β :x−→ 2x1(F7)
ψ(β.∞) = β.P.β−1=P=ψ(∞)
k∈F7
ψ(β.k) = β(αkQα−k)β−1= (βαkβ−1)(βQβ−1)(βα−kβ−1) = α2kQα−2k=ψ(2k)
Q=<2>={1,2,4} ⊂ F?
7F? 7N=F? 7\Q=
{3,5,6}β1(F7) 0 ∞Q N
β(1,2,4)(3,6,5)
M G 1< n3= 3k+ 1|56
n3

n0
3G(P)n0 3= 3k+ 1|7
n0
3G(P)n0 3= 7
n3= 28
G(M)G(M)C6S3
U G
v−→< v2> G
U6=∅u∈U H G g ∈G
H=g < u2> g−1H=<(gug−1)2> gug−1
v−→ v2U G
w∈G < w > < w >=< v2>
v w =v2w=v4= (v5)2v5G
n7= 8 n3= 28
G
U=∅
G(M)'S3
G(M) =< β, γ > β γ γβγ−1=β−1
G(M)γ γ0=βγβ−1γ00 =β2γβ−2
G(P)G H G
G(P)H
G H
φH:G−→ SG/H
g−→ φH(g) : x−→ gx
168|4!
G
G=< α, β, γ >
GΩ1(F7)G∞=G(P) =< α, β >
γ(∞)6=∞β−1γ(∞) = γβ(∞) = γ(∞)γ(∞)β
γ(∞)=0 γ∞
γ0∞β k ∈1(F7)\{0,∞} =
F?
7
β−1γ(k) = γβ(k) = γ(2k)=4γ(k)
γ(1) ∈Q γ(Q) = Q γ (1,2,4)
γ(1) ∈N γ Q N β γ
γ0γ00 G(P)γ(1) = 6
γ(2) = 3 γ(4) = 5 γ= (1,6)(2,3)(4,5)(0,∞)γ
x−→ − 1
x1(F7)
G'2(F7)
X
D ⊂ P(P)X
x, y ∈P D ∈ D x, y ∈D
D, D0∈ D D∩D0={x}x∈P

X
D∈ D
X={1,2,3,4,5,6,7}
D={{1,2,4},{2,3,5},{3,4,6},{4,5,7},{5,6,1},{6,7,2},{7,3,1}}
G(X, D)
G={σ∈S7/D ∈ D ⇒ σ(D)∈ D}
π= (1,2,3,4,5,6,7) ∈G G α = (2,7,6)(4,3,5) ∈G β =
(2,4)(5,6) ∈G1G
σ∈G1,2σ(4) = 4
G1,2={e, (3,5)(6,7),(3,6)(5,7),(3,7,)(5,6)}
(G)=7.6.(G1,2) = 168
K G G K
G G
n7K G
(1,7,5,3,4,2,6) = απα−1∈G π
n7= 8 (K)≥49 K=G G
G X B ⊂X
X g ∈G g(B) = B g(B)∩B=∅ ∅
X{x}x∈X G
X k ≥1X(k)⊂Xkk
(x1, . . . , xk)G X
k G X(k)G X
G X k k ≥2
OB x y z ∈X z 6=x
g∈G g(x) = x g(y) = z g(B) = B z ∈B B =XM
G X
K G x
OKM