Décomposition cyclique d`un groupe abélien fini

G n ≥2q1
q2q1qkqk−1
G(Z/q1Z)×...×(Z/qkZ)
G a ∈G o(a)y∈G/ < a >
x∈G x =y o(x) = o(y)
G ϕ
G G/ < a > s y ∈G/ < a > x ∈G ϕ(x) = y
ϕ(sx) = sy = 0 sx ∈(ϕ) =< a > k ∈N0≤k < o(a)sx =ka
k=sq +r0≤r < s
sx =ka =sqa +ra x0=x−qa
ϕ(x0) = ϕ(x) = y s =o(y)|o(x0)
r6= 0 sx0=sx −sqa =ra o(sx0) = o(x0)
o(x0)∧so(sx0) = o(x0) s
o(x0) = so(sx0) = so(ra) = so(a)
o(a)∧r
o(a)o(x0)≤o(a)s≤o(a)∧r≤r
r= 0 sx0=ra = 0
o(x0)|s o(x0) = o(y)
n G n =p
GZ/pZ
n G n a ∈G m m > 1G6={e}
G/ < a > |G|=n
G/ < a > G0
1=< a0 1> G0
k−1=< a0 k−1> q1qk−1
1< q1|...|qk−1G/<a>=G0
1×. . . ×G0
k−1G
a1ak−1a0
1a0
k−1G
G1=< a1> Gk−1=< ak−1> Gk=< a >
ϕ G G/ < a > x ∈G n1nk−1
0≤n1< o(ai)i= 1, . . . , k −1
ϕ(x) = n1a0
1+. . . +nk−1a0 k−1=ϕ(n1a1+. . . +nk−1ak−1)
nka(ϕ) =< a > 0≤nk< m =o(a)x= (n1a1+. . . +
nk−1ak−1) + nka G =G1+. . . +Gk
x G
G1, . . . , Gkx=m1a1+...+mka x ϕ(x) =
n1a0
1+. . .+nk−1a0
k−1=m1a0
1+. . .+mk−1a0
k−1G/ < a > G0
1, . . . , G0
k−1
n1=m1, . . . , nk−1=mk−1nka=mka nk=mk0≤nk< o(a)
0≤mk< o(a)
x0= (a1, . . . , ak−1, a)∈G1×. . . ×Gko(a1)o(a)
o(x0)≥o(a)o(a)G o(a) =
o(x0) = ppcm (o(a1), . . . , o(ak−1), o(a)) o(a1)|. . . |o(ak−1)|o(a)
q1, . . . , qkn G
n=p(qi)p
n G n > 1
G=G1×. . . ×Gk=G0
1×. . . ×G0 m
Gi'Z/qiZG0
j'Z/q0 jZ1< q1|...|qk1< q0
1|. . . |q0
m
p q1q2, . . . , qkG f :x7→ px
G Gi

f(G1)⊂G1G1
q1
pf(G2)⊂G2f(Gk)⊂Gk
q1
p, . . . , qk p
(f(G1) + . . . +f(Gi)) ∩f(Gi+1)⊂(G1+. . . +Gi)∩Gi+1 ={0}
i= 1, . . . , k −1f(G)f(G1), . . . , f(Gk)|f(G)|=q1. . . qk
pk=|G| pk
f(G0
j)⊂G0 jj= 1 . . . m f(G) = f(G0
1)×...×f(G0
m)q0
1|...|q0
m
r p q0
1, . . . , q0
rp q0
r+1, . . . , q0
mf(G0
1) = G0 1f:x7→ px
G0
1f(G0 2) = G0 2f(G0
r) = G0 r
f(G0
r+1), . . . , f (G0
m)q0
r+1
p, . . . , q0
m
p|f(G)|=q0
1. . . q0 r
q0
r+1 . . . q0
m
pm−r=|G| pm−r
f(G)k=m−r≤m
m≤k m =k r = 0 p q0
1, . . . , q0
k
f(G)q1
p,...,qk p
q0
1
p, . . . , q0
k
pq1, . . . , qkq0
1, . . . , q0
k