Corrigé d'exercices d'algèbre : Groupes et automorphismes

(E, ·)a∈E
Ga, Da:E→E Ga(x) = ax Da(x) = xa x ∈E
a GaDa
(E, ·)E
(E, ·)
a∈E GaDa
(G, ·)H G ·H
G
(E, ·)
a x, y ∈E Ga(x) = Ga(y)ax =ay a
x=y Ga
Gax, y ∈E ax =ay Ga(x) = Ga(y)
x=y Ga
a Ga
a Da
(E, ·)a∈E a
GaDaE∃e∈E
ae =Ga(e) = a x ∈E Dax0∈E x =x0a xe =
(x0a)e=x0(ae) = x0a=x a(ex)=(ae)x=ax a ex =x
∀x∈E, xe =ex =x(E, ·)e
(E, ·)e x ∈E x0, x00 ∈E Gx(x0) = xx0=e
x00 x=Dx(x00 ) = e E (E, ·)
(G, ·)H G ·(H, .)
H G
E≥2E∗x∗y=y∗
E a ∗x=a∗y⇒x=y(E, ∗)
nUn(Z/nZ,·)
¯
k∈Z/nZ
¯
k∈Un
k n
¯
k(Z/nZ,+)
φ(n)Unφ(n)n
φ(n) = card{k: 1 ≤k≤n, k ∧n= 1}
p s φ(ps) = ps−ps−1
m n
f:Z/mnZ→Z/mZ×Z/nZ
¯x[mn]7→ (¯x[m],¯x[n])
¯x[k]x k
card(Umn) = card(Um×Un)φ(mn) = φ(m)φ(n)
n=pk1
1···pks
sn piφ(n)
piki
(i)⇒(ii)¯ k∈Unm∈Z¯ k¯m=¯ 1n|km −1
s∈Zkm −1 = sn km −sn = 1 k m
(ii)⇒(iii)k∧n= 1 m, s ∈Zkm +sn = 1 m¯ k=¯ 1
∀t∈Z¯ t=tm¯ k¯ k(Z/nZ,+)
(iii)⇒(i)¯ k(Z/nZ,+) m∈Zm¯ k= 1 ¯m¯ k=¯ 1

¯
k
φ(ps)k1≤k < ps
psE={1,2, . . . , ps}F={k∈E:p|k}φ(ps) = card(E)−card(F) =
ps−card(F)F={p, 2p, 3p, . . . , ps=ps−1p}(F) = ps−1φ(ps) = ps−ps−1
m n
f:Z/mnZ→Z/mZ×Z/nZ
¯x[mn]7→ (¯x[m],¯x[n])
f¯x[mn] = ¯y[mn]mn |x−y m n
m|x−y n |x−y¯x[m] = ¯y[m] ¯x[n] = ¯y[n]
f f(¯x¯y[mn]) = (xy[m], xy[n]) == (x[m], x[n])(y[m], y[n]) =
f(¯x[mn])f(¯y[mn])
f(¯x[m],¯x[n]) = (¯y[m],¯y[n]) m|x−y n |x−y m n
mn |x−y¯x[mn] = ¯y[mn]f
¯x∈Umn ⇔f(¯x)∈U(Z/mZ×Z/nZ=Um×Unφ(mn) = card(Umn =card(Um×Un) =
card(Um)(Un) = φ(m)φ(n)
n=pk1
1···pks
sφ(n) = Qs
i=1 φ(pki
i) = Qs
i=1(pki
i−pki−1 i)
Gn={z∈C|zn= 1}
Gn(C∗,·)
G n (C∗,·)G=Gn
G
Gn(C∗,×) 1 ∈Gnu, v ∈Gn(uv−1)n=
un(vn)−1= 1 uv−1∈Gn
Gnz=exp(2kπi
n) = exp(2πi n)kGn=grhξi={1, ξ, ξ2, . . . , ξn−1}
ξ=exp(2πi
n)
G n (C∗,×)∀z∈G zn= 1
G⊂Gn|G|=|Gn|G=Gn
z=a+ib ∈Ca, b ∈R(z) = ea(cos b+isin b)
f: (C,+) →(C∗,×)f(z) = exp(z)
u, v ∈Cu=a+bi v =c+di a, b, c, d ∈R
exp(u+v) = exp(a+c)(cos(b+d) + isin(b+d)) = exp(a) exp(c)[cos(b) cos(d)−sin(b) sin(d) + isin(b) cos(d) +
isin(d) cos(b)]
exp(u+v) = exp(a)(cos(b) + isin(b)) exp(c)(cos(d) + isin(d)) = exp(u) exp(v)
exp (C,+) →(C∗,×)
z=a+bi ∈Cu∈Kerf ⇔exp(z) = ea(cos(b) + isin(b)=1 ea(cos(b)=1 ea(sin(b)=0
ea>0 sin(b) = 0 cos(b) = 1 a= 0 b= 2kπ k ∈Z
Kerf ⊂2πiZz= 2kπi k ∈Zexp(z) = 1
Kerf = 2πiZ
z=a+bi ∈Imf z 6= 0 z=ρeiθ u=c+di exp(u) = ecedi =ρeiθ
c= ln(ρ)d=θ2π z ∀u∈C∗f
f=C∗
G x 7→ x−1G G
∀x, y ∈G(xy)−1=y−1x−1= (yx)−1xy =yx G
2(R)A=0 1
−1−1B=0−1
1 0 A, B
AB

A2=−1−1
1 0 A3=I o(A) = 3 B2=−I B3=−B B4=I o(B) = 4
AB =1 0
−1 1 AB =I+N N =0 0 −1 0 (AB)k= (I+N)k=I+kN 6=I, ∀k∈N∗AB
G g ∈G γg:G→G γg(x) =
gxg−1,∀x∈G
γgG g
γgh =γg◦γh(γg)−1=γg−1
(G)G(G)
G
(G) (G)
(G)∼
=G/Z(G)Z(G)G
∀x, y ∈G γg(xy) = gxyg−1=gxg−1gyg−1=γg(x)γg(y)γg
γg◦γg−1=γg−1◦γg=IGγg
∀x∈G γgh(x) = ghx(gh)−1=ghxh−1g−1=γg(γh(x)) = γg◦γh(x)γgh =γg◦γh
γg◦γg−1=γe=IGγg−1= (γg)−1
IG∈(G)γg, γ−1
h∈(G)γg◦γ−1 h=γgh−1∈(G) (G)
(G)
σ∈(G)∀x∈G σ ◦γg◦σ−1(x) = σ(gσ−1(x)g−1) = σ(g)xσ(g−1) =
γσ(g)(x)σ◦γg◦σ−1== γσ(g)∈(G) (G)
(G)
φ:G→(G)φ(g) = γgφ(gh) = γgh =γg◦γh=φ(g)◦φ(h)φ
G/ φ ∼
=(G)g∈G
g∈φ⇔γg=IG⇔ ∀x∈G, gxg−1=x⇔ ∀x∈G, gx =gx ⇔g∈Z(G)
f:G→G0x∈G f(x)
x f
G x, y ∈G xy yx
xyz, yzx, zxy
n=o(x)f(x)n=f(xn) = f(e) = e0o(f(x))|n
f k ∈Nf(x)k=e0f(xk) = e0f xk=e
n|k o(f(x)) = n
xy =xyxx−1=γx(yx)γxx
o(xy) = o(γx(yx)) = o(yx)
xyz =xyzxx−1=z−1zxyz
m n
Z/n.Z×Z/m.Z∼
=Z/nm.Z
φ:Z→Z/n.Z×Z/m.Zφ(x) = (π1(x), π2(x)) π1(x)π2(x)
x n m φ
x∈Zx∈(φ)⇔n|x m|x⇔mn|x m n
φ=nmZ Z/ φ =Z/nmZ∼
=φ
o(φ) = o(Z/nmZ) = mn =o(Z/n.Z×Z/m.Z)φ=Z/n.Z×Z/m.Z
Z/n.Z×Z/m.Z∼
=Z/nm.Z
G1G2n m G1×G2∼
=
Z/n.Z×Z/m.Z∼
=Z/nm.ZG1×G2

G H G
g∈G gHg−1G H |gHg−1|=|H|
G H m H G
g∈G γg:G→G γg(x) = gxg−1
gHg−1=γg(H)H
γg|gHg−1|=|γg(H)|=|H|
G H m g ∈G|gHg−1|=|H|=m
gHg−1=H H CG
G e ∀x∈G x2=e
G
G2
x, y ∈G e = (xy)2=xyxy xy =x2yxy2=yx G
p p | |G|
G x p xp=e2|p p = 2 2
|G| |G|2
G H G
H G
∀g∈G g2∈H
a−1Ha ⊂H a ∈G a ∈H a /∈H
[G:H] = 2 H(G/H)g={H, Ha}
H(G/H)d={H, aH}G=H∪Ha =H∪aH Ha =G\H=aH
a−1Ha ⊂H
g∈G g ∈H g2∈H g /∈H g2/∈H g2∈Hg G =H∪Hg
g2=hg h ∈H g =h∈H g2∈H
G={e, a, b, c}e
a2=b2=c2=e ab =ba =c ac =ca =b bc =cb =a G
G
≤5
x∈G o(x)||G|= 4 o(x)∈ {1,2,4}G o(x) = 1 2
x2=e
ab ba 6=a, b ab =ba =c ac =ca =b bc =cb =a G
H={e, a}K={e, b}HK =G H ∩K={e}H, K CG G
G∼
=H×K
G≤5
• |G|= 1 {e}
• |G=|p= 2,3,5 (Z/pZ,+) p
• |G|= 4 (Z/4Z,+) (Z/2Z×Z/2Z,+)
G H G E G
H E = (G/H)g={xH :x∈G} B(E)E
a∈G ρa:E→E ρa(xH) = axH
ρaρa∈ B(E)

GΦ : G→ B(E)a7→ ρa
N
G/N B(E)
N G H
H m
[H:N] (m−1)!
m|G|H
G
a∈G xH =yH x−1y∈H(ax)−1(ay) = x−1a−1ay =x−1y∈H
axH =ayH ρa
G×E→E(g, xH)7→ gxH G E
xH yH ρa(xH) = ρa(yH)axH =ayH (ax)−1(ay)∈H(ax)−1(ay) = x−1y
xH =yH ρa
yH ∈E ρa(a−1yH) = yH ρa
ρaE
∀a, b, x ∈G ρab(xH) = abxH =ρa(ρb(xH)) ρab =ρa◦ρbΦ(ab) = Φ(a)◦Φ(b) Φ
G/ Φ∼
=Φ
B(E)G/N B(E)
N G g ∈N
gxH =H x ∈G x =e gH =H g ∈H N ⊂H
K G H K ⊂N g ∈K x ∈G
KCG x−1gx ∈K K ⊂H x−1gxH =H gxH =xH
g∈N
G/N B(E)E= [G:H]B(E)∼
=Sm
|G/N|= [G:N] = [G:H][H:N]|m! [H:N]|(m−1)!
[H:N] = 1 p[H:N]p|(m−1)!
p k = 1, . . . , m −1p≤m−1< m p
|G|p≥m m |G|
G pq p q p < q
G H p N q
G G
G H p N q
NCG G =NH N q G
2q q 6= 2
p q G
G a q b p N =grhaiH=grhbi
G HN G |HN |=|H||N|/|H∩N| |H| |N|
H∩N={e} |HN |=|H||N|=pq =|G|G=HN ∼
=H×N
G
N p G
N G G =HN
p= 2 G G
G=HN =grha, bio(a) = q o(b) = 2 (ab)2=abab =e bab =a−1
NCG bab =bab−1=ak∈N a =bbabb =bakb= (bab)k= (ak)k=ak2
ak2−1=e q |k2−1 = (k−1)(k+ 1) q q |k−1q|k+ 1
k∈ {0,1,2. . . q −1}k= 1, k =p−1k= 1 ab =ba G
k=p−1bab =ap−1=a−1G
(G, ·)e G 6={e} {e}G
G
G
G