Groupes et Géométrie

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p2G G G
g∈G cg:G→G:x7→ gxg−1
G x, y ∈G
g∈G cg(x) = gxg−1=y
x∈G CG,x ={g∈G|cg(x) = x}
G Z(G) = {g∈G| ∀x∈G:cg(x) = x}
CG,x G[G:CG,x]
x G G
G→Aut(G)
G G/Z(G)
G G =Z(G)G/Z(G) = {eG/Z(G)}
G G
G p G
p p
p2p3
p3
Sn
σ∈Snσ= (i1i2i3· · · ik)τ∈Sn
τστ−1(τ(i1)τ(i2)τ(i3)· · · τ(ik))
σ1, σ2∈Sn
σ1σ2
σ1, σ2∈Snσ1
σ2σ1σ2
Snn
n=k1+k2+· · · +ks0< k1≤k2≤ · · · ≤ ks
S2S3S4
G N H N ∩H={eG}NH =G
x∈G x =nh n ∈N h ∈H
nh =hn n ∈N h ∈H G ∼
=N×H
G N H
G pq p, q G
p q G
2 3 (Z/6Z,+) S3
G N, H N N ∩H={eG}NH =G
x∈G x =nh n ∈N h ∈H
φ:G→H nh ∈G h ∈HKer(φ) = N
G N H G ∼
=NoH
ρ:H→Aut(N)h∈H
ρh:N→N:n7→ hnh−1x1=n1h1x2=n2h2x1x2=n1ρh1 (n2)h1h2
S3∼
=A3oS2ρ:S2→Aut(A3)A3S2
D2n∼
=(Z/nZ,+) oS2ρ:S2→Aut(Z/nZ,+)
2S2kGSm
G1G2S2mG G1G2
m+1, m+2,...,2m1,2, . . . , m G1∩G2={eS2m}G1
G2
S2oGS2mG1G2τ∈S2mτ(i) = i+m
i≤m τ(i) = i−m i > m S2oG(G1×G2)oS2
S2→Aut(G1×G2)S2oG2n2n G
21
22m mG2SmS2oG2S2m
m= 2k
k2S2k
p