Petits groupes 1 Groupes d`ordre p premier 2 Groupes d`ordre p2, p

<16
p
1
pZ/pZ
p
Z/pZ
2,3,5,7,11,13
p2p
G p2
p p
p
G
Z(G) = {g∈G:∀h∈Ghg =gh}={g∈G:∀h∈Ghgh−1=g}.
p G
G
G1
p
G p p G
G p2Z(G)
p
p x G
Z(G)x G
Z(G)x
Z(G)Z(G)
G x G
G p2Z/p2Z

p G
Z/pZ(Z/pZ)2
4,9
pq p q p < q
q N =qαr q 6 −r1q
r
G|G|=pq
q1q p
1q H G q H q
Z/qZK p G p
Z/pZH∩K={1}H
q K p
|G|=|H| |K|G H K
G=HoK
GZ/qZoZ/pZ
Z/pZ Z/qZ
ρ:Z/pZ−→ Aut(Z/qZ)'(Z/qZ)×'Z/(q−1)Z.
ρ G Z/qZ×
Z/pZ'Z/pqZp-(q−1)
15 = 3 ×5 15 Z/15Z
(q−1) = pr ρ
Z/(q−1)Zr
ρ
ρZ/pZ
pq Z/qZ×Z/pZ'Z/pqZ
Z/qZoZ/pZ
p= 2
p= 3 21 = 3 ×7Z/2Z
G2G ρ
2Z/2Z Z/2Z Z/qZ
m7→ −m
Dq2q q
Z/2qZDq
6 10
8
Z/8Z Z/4Z×
Z/2Z(Z/2Z)3G8
G8Z/8Z

G2G
xy = (xy)−1=y−1x−1=yx G
(Z/2Z)3
G i 4 8
i2
G/C Z/2Z
1−→ Z/4Z−→ G−→ Z/2Z−→ 1.
G\CZ/2Z
GZ/4ZoZ/2Z
ρ:Z/2Z→Aut(Z/4Z)'(Z/4Z)×'Z/2Z Z/2Z
Z/4Z Z/4Z×Z/2Z
−1Z/4ZoZ/2Z
D4
G\C
4j j2C2j2=i2
ji 6=ij ij 2G\C G
2
2i2G2Z={1, i2}
i2=−1C1, i, −1,−i G \C j, ij, −j, −ij
ji G \C j, ij, −j=j(−1) ji =−ij
G i j i2=j2=−1ji =−ij
H8
8
Z/8Z,Z/4Z×Z/2Z,(Z/2Z)3, D4,H8.
12
Z/12Z Z/6Z×Z/2Z
G1 4 3
1H G
Z/3Z2K G H ∩K={1} |H| |K|=|G|
G=HoK K Z/4Z Z/2Z×Z/2Z
Z/3Z Z/2Z
K'Z/4Z Z/4Z Z/3Z
Z/4Z→Z/2Z
Z/3Z o Z/4Za, b a3=
b4= 1, ba =a2b
K'(Z/2Z)2Z/2Z×Z/2Z→
Z/2Z
(Z/2Z)2
Z/2Z
Z/3Z o (Z/2Z)2
a, b, c a3=b2=c2ba =ab cb =bc

ca =a2c u =ab u, c
u6=c2= 1 cu =u5c
D6
4 3 G
3{1}
3 1 8 3 3
1 2 K
G3H K ∩H={1} |K| |H|=|G|
G=KoH H Z/3ZKZ/4Z
(Z/2Z)2
K'Z/4Z
Z/2Z Z/3Z Z/4Z
Z/4Z×Z/3Z'Z/12Z
K'(Z/2Z)2GL(2,Z/2Z)' S3
Z/3Z→ S33
3S3
(Z/2Z)2o Z/3Z
4 3
G3
ϕ:G→ S43
H H 3G
H ϕ
3{1}ϕ G
S42S4
A4
S4→Z/2Z
D6
D6Z/12Z
Z/6Z×Z/2Z6A4
4cycl3o Z/4Z
Z/3ZoZ/2Z2

Z/2Z
Z/3Z
Z/4Z(Z/2Z)2
Z/5Z
Z/6ZD3' S3
Z/7Z
Z/8Z Z/4Z×Z/2Z(Z/2Z)3D4H8
Z/9Z(Z/3Z)2
Z/10ZD5
Z/11Z
Z/12Z Z/6Z×Z/2ZD6A4
Z/3ZoZ/4Z' ha, b |a3=b4= 1, ba =a2bi
Z/13Z
Z/14ZD7
Z/15Z