2 Groupes monogènes, groupes cycliques. Exemples

Z
nZ
(G, ·).
(Hi)i∈IG
G.
H=
i∈I
Hi.1Hi,
H H ̸=∅. g1, g2H, Hi, g1g−1
2∈Hi
i∈I, g1g−1
2∈H. H G.
X(G, ·), G
X G.
G X
G
G.
X(G, ·), G X
G X.
⟨X⟩G X ⟨X⟩
G X.
X⟨X⟩={1}.
X G x1,··· , xn,
⟨X⟩=⟨x1,···, xn⟩X.
X(G, ·), X G X
G G =⟨X⟩.

G
X, Y G.
X⊂ ⟨X⟩X G.
X⊂Y, ⟨X⟩ ⊂ ⟨Y⟩.
X X−1X,
X−1={x−1|x∈X},⟨X⟩x1·····xrr∈N∗
xkX∪X−1k1r.
H=x1· ··· · xr|r∈N∗xk∈X∪X−11≤k≤r
G.
x1∈X, 1 = x1·x−1
1∈H x =x1· ··· · xr, y =y1· ··· · ysH,
x·y−1=x1· ··· · xr·y−1
s· ··· · y−1 1∈H
H G.
H G X, ⟨X⟩ ⊂ H.
h=x1···· ·xrH X ∪X−1⊂ ⟨X⟩,⟨X⟩
⟨X⟩=H.
⟨X⟩=⟨X−1⟩=⟨X∪X−1⟩.
⟨X⟩=r
k=1
xεk
k|r∈N∗, xk∈X εk∈ {−1,1}1≤k≤r
X
n(g1,··· , gp)G
p≥1
⟨g1,··· , gp⟩=p
k=1
gαk
k|(α1,··· , αp)∈Zp
⟨g1,··· , gp⟩
X={g1,··· , gp}, X−1=g−1
1,··· , g−1
p
gk
⟨g1,··· , gp⟩=m
k=1
hk|m∈N∗hk∈X∪X−11≤k≤m
=p
k=1
gαk
k|(α1,··· , αp)∈Zp
gkgj=gjgkg−1
jgk=g−1 jgkgjg−1 j=g−1 jgjgkg−1 j=gkg−1 jX∪X−1

gk
G
⟨g1,··· , gp⟩=p
k=1
αkgk|(α1,··· , αp)∈Zp
G=Z,
⟨g1,···, gp⟩=
p
k=1
gkZ=δZ
δ∈Npgcd g1,··· , gp.
G g ∈G G =⟨g⟩. G
⟨g⟩=r
k=1
gεk|r∈N∗, εk=±1 1 ≤k≤r={gn|n∈Z}
⟨g⟩={ng |n∈Z}
g∈G⟨g⟩g
⟨g⟩. θ (g)
(θ(g) = n∈N∗)⇔(⟨g⟩={gr|0≤r≤n−1})
⇔k∈Zgk= 1 k≡0 mod (n)
n=θ(g)gn= 1 ng = 0
G g ∈G
G.
g̸= 1
1g.
(Z,+) 1.(Z,+)
nZn≥0
GZ, nZn
(Z,+) Z
nZ
k
k=qn +r0≤r≤n−1, k −r∈nZk=r. r ̸=s

0≤r̸=s≤n−1 0 <|r−s|< n r −s n
Z
nZ=0,1,··· , n −1
n n 1.
Z
nZ
X={r1,··· , rn}QG=⟨X⟩(Q,+)
X. G
µppcm r1,··· , rn,
a1,··· , anrk=ak
µk1n δ
pgcd a1,··· , an,
G=n
k=1
αk
ak
µ|(α1,··· , αn)∈Zn
=δ
µ
n
k=1
αkbk|(α1,··· , αn)∈Zn
b1,··· , bnG=δ
µZ,
Gδ
µ.
G
Z.
n, Z
nZ.
G=⟨g⟩φg:k7→ gk
(Z,+) GZ,
ker (φg) = nZn∈N.
n= 0, φgGZ.
n≥1,Z
nZG G =⟨g⟩
n.
G n, G n
G g n.
G=⟨g⟩=1, g, ···, gn−1
G=⟨g⟩n. G gk,
k1n−1n.
k∈ {1,··· , n −1}n,
u, v uk +vn = 1, g =gku∈gk
G=gk.

k∈ {1,··· , n −1}G=gk, u
g=gku=gku, g1−ku = 1 n ku n g
v1−ku =vn, uk +vn = 1 k
n.
n, φ (n),1n n
n= 1, φ (1) = 1
G n φ (n).
n≥2φ(n),
n
m= ppcm {θ(g)|g∈G}=θ(g0)n
m=
r
k=1
pαk
kn=
r
k=1
pβk
k, pk1≤αk≤βkk
1n.
φ(n) =
r
k=1
pβk−1
k(pk−1)
φ(n)n, βk1pkφ(n)n
αk1, n =m G g0n= card (G).
n≥2
φ(n), n
Γnn
n, Z
nZk7→ e2ikπ
n.
n≥1,(Q/Z,+)
n
G(Q/Z,+) n. r ∈G
n nr = 0, q ∈Z
nr =q r =q
n. r =q n=q1 n∈1 nG⊂1 n.1 nn
Q/Zk1
n=k n= 0 k n∈Z, k
n G =1
n. n
1
n
p
Z
pZ.