149: groupes finis de petit cardinal

GZ/nZ
Zn
Z/nZ
p1, . . . , pn
Z/p1p2. . . pn'Z/p1Z×Z/p2Z×. . . ×Z/pnZ Z/p1p2. . . pn'
Z/p1Z×Z/p2Z×. . . ×Z/pnZp1, . . . , pn
n=Qpαi
iPαi
A(2) = {Z2}A(3) = {Z3}A(4) = Z2
2, Z4A(8) = Z3 2, Z2×Z4, Z2×Z2×Z2
A(10) = {Z2×Z5}A(24) = Z3×Z8, Z3×Z2×Z4, Z3×Z3
2A(32) = Z4 2, Z2×Z16, Z4×Z8, Z2×Z2×Z8,
A(48) = {Z3×Z16, Z2×Z3×Z8, Z3×Z4×Z4, Z2×Z2×Z3×Z4}
p|n
Znd d|n n/d
d d
n Za1×. . . ×Zai
p
A4
Zpp
Z2Z3Z5Z7Z11 . . . Z59
p1, . . . , pn
(Z/p1p2. . . pn)∗'Z/p1Z∗×Z/p2Z∗×. . . ×Z/pnZ∗
Auth(Zn)'Zn∗n=Qpαi
ipi
p1= 2, α1>2Aut(Zn)'Z2×Zα−2
2×Pα−1 2(p2−1) ×. . . ×pαn
n(pn−1)

p1= 2, α1= 1 Aut(Zn)'pα−1
2(p2−1) ×. . . ×pαn
n(pn−1)
p1>3Aut(Zn)'pα−1
2(p2−1) ×. . . ×pαn
n(pn−1)
Aut(Z2) = {Z1}Aut(Z3) = {Z2}Aut(Z4) = {Z2}Aut(Z10) = {Z4}Aut(Z16) =
{Z2×Z4}Aut(Z20) = {Z2×Z4}Aut(Z33) = {Z2×Z2×Z3}
Aut((Zp)n'GLn(Fp))
Aut(V4)'GL2(F4) = S3Aut(Z2
3) = Gl2(F3)
Aut(Z2×Z4) = D4.
|G|=pα∗m, p pgcd(p, m)=1
pα
pα
N H ϕ
N×H(n, h)(n0, h0)=(nϕ(h)n0, hh0)N×H
N H N oϕH
N N oϕH H
NoϕH⇔ϕ⇔
N G H G NH
G H N N G NH =G N ∩H= 1
G
Z/ZD4
H8
p p2
p Zp
NA(1) = NA(3) = . . . =NA(59)
p2Z/p2Z Z/pZ2
Z/pZ×Z/qZ
Z/pZ×Z/qZ
Z/pZoZ/qZ
S3Z/3Z

p2q
p2q
C1p q −1C2p2q−1
C3q p −1C4q p + 1 C2C1
q= 2 Zp2oZ2Z2
po1Z2Z2 po2Z2
q>3
C1Zp2oZqZ2
poZq
C2ZqoZ2
pZqo1Zp2Zqo2Zp2
C2C3ZqoZp2Zqo1Z2
pZqo2Z2 pZp2oZq
C2C4ZqoZp2Zqo1Z2
pZqo2Z2 pZ2 poZq
C3Zp2oq
C4Z2
poq
p3
p3Z/p3Z Z/pZ2×
Z/Z2Z/p3Z Z/pZ2o Z/Z2Z/pZ2o Z/Z
NA(8) = D4, H8
NA(30) = {D15, Z3oD5, Z5oD3}
NA(36) = {Z21o1Z2, Z21o2Z2, Z21o3Z2, Z7o1Z6, Z7o2Z6}
NA(42) = Z2
2oZ9, Z2 2oZ2 3, Z9oZ4, Z9oZ2 2, Z2 3oiZ2 2, Z2 3oiZ4i= 1,2,3
A5
16,24,32,40,48,56,60
Zni=
8,16,32,40,48,56
A5
60
Zp

Ann= 3 n>5
SnAnD(Sn) = D(An) = An
Int(Sn)'Aut(Sn)'Snn= 6 Aut(S6)'S6o2
Sn'Ano2
S3
S4
A4