Contrôle continu n 1

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G, G0f:G→G0
Im(f)Ker(f)G G0
Im(f)G Ker(f)G0
|G|=|Im(f)| |Ker(f)|.
f1f1:G→Im(f)f1
¯
f1:G/Ker(f)→
Im(f)Ker(f)/Ker(f)Im(f)
G/Ker(f)
|Im(f)|=|G/Ker(f)|=|G|/|Ker(f)|,
G f G G0a
G f(a)Im(f)y∈Im(f)x∈G f(x) = y
G n ∈Zx=any=f(a)n
GRG
gRg0x∈G g =xg0x−1
RG g g0
G G
Rg∈G g =ege−1gRgR
g, g0∈G gRg0x∈G g =xg0x−1
x−1gx =g0
g0= (x−1)g(x−1)−1,

g0RgRg, g0, g00 gRg0g0Rg00
x, y g =xg0x−1g0=yg00 y−1
g=xyg00y−1x−1= (xy)g00 (xy)−1,
gRg00
H G f :G→ {0,1}
H f(x) = 1 x∈H f(x) = 0 f
R¯ f:G/R → {0,1}
gRg0
f(g) = f(g0)g, g0x g =xg0x−1g0H
g H g g0H
H f(g) = f(g0)
x∈G cx:G→G cx(g) = xgx−1
g, g0∈G
cx(g)cx(g0) = xgx−1xg0x−1=x(gg0)x−1=cx(gg0),
cxcx◦cx−1=IdGcx−1◦cx=IdG
cx
g, g0g g0
cx
x g =xg0x−1=cx(g0)n cx(g0n) = gn
cxg0hg0i={g0n:n∈Z}
g cxGhg0i
hg0i hgi
o:G→NgR
g, g0
g g0
g g0o(g) = o(g0)