Le lemme qui n`est pas de Burnside

G E
·G×E E (gh)·x=g·(h·x)
eG·x=x x E g h G G
E E g ·x=g(x)
G E
g∈G
g(x) = x x g G
E x ∼y⇐⇒ ∃g∈G, g(x) = y
G
G E E/G
FggCard(E/G)Card(G) = P
g∈G
Card(Fg)
(E/G)×G
F={(x, g)∈E×G|g(x) = x}
f(x) = g(x) = y x y
gf−1(y) = y y gf−1
xC∈E C ∈E/G
E y ∈C gy∈G gy(xC) = y
(C, f)∈(E/G)×G gf(xC)(xc) = f(xC)y=f(xC)
h=gf(xC)f−1ϕ((C, f)) = (y, h)ϕ
(E/G)×G F ϕ
ϕ((C, f)) = ϕ((C0, f0)) = (y, h)C
y C =C0gf(xC)f−1=gf(xC)f0−1f=f0
ϕ h ∈G h(y) = y Cy
y ϕ((Cy, h−1gy)) = (y, h)ϕ
Card((E/G)×G) = Card(E/G)Card(G)
F=S
g∈G
Fg× {g}
n
E
{A, B, C, D}n
{1,2, . . . , n}E={1,2, . . . , n}{A,B,C,D}f∼g⇐⇒
∃s, s , g(S) = f(s(S)) S∈ {A, B, C, D}
E G
s·f=g
s(f) = g g g(X) = f(s(X)) X∈ {A, B, C, D}
G kπ/2
E
n4n2
n3n
x=n4+ 2n3+ 3n2+ 2n
8=n(n+ 1)(n2+n+ 2) 8.