Classification des groupes d`ordre pq

G pq p < q
q1p G Z/pqZ
q1p G
GZ/pqZ
G G (Z/qZ)oθ(Z/pZ)
θ∈Hom(Z/pZ, Aut(Z/qZ)) θ(1) = γ p Aut(Z/qZ)
G H q K
p nqq1q p
p < q nq= 1 H G
|H∩K| |H|=q|K|=p|H∩K|= 1
H∩K={e}HCK HK G HK/H '
K/(H∩K) = K|HK|=|H||K|=pq =|G|HK =G G
H K (Z/qZ)oθ(Z/pZ)θZ/pZ
Aut(Z/qZ)
p K G np1
p q np= 1 np=q np=q q 1p
q1p np= 1 K
G H ×K p
q H K G
Z/pqZ
q1p θ :Z/pZ→Aut(Z/qZ)p
Z/pZp1
θ(Z/qZ)oθ(Z/pZ)
1. G Z/pqZ
θ Aut(Z/qZ)
ϕ(q) = q−1p Aut(Z/qZ) Γ p
Γ = Im(θ)Z/pZΓ = Im(θ)θ
Z/pZΓθ(1) = γ p −1 Γ
p−1θ(1) θ0
Z/pZΓα=θ0−1◦θZ/pZ
f GθGθ0
p= 2 q > 2G2qZ/2qZ
Dp
G G 'Z/2qZ
Dq2q