(E, ·)aE
Ga, Da:EE Ga(x) = ax Da(x) = xa x E
a GaDa
(E, ·)E
(E, ·)
aE 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, ·)aE a
GaDaEeE
ae =Ga(e) = a x E Dax0E x =x0a xe =
(x0a)e=x0(ae) = x0a=x a(ex)=(ae)x=ax a ex =x
xE, 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
E2Exy=y
E a x=ayx=y(E, )
nUn(Z/nZ,·)
¯
kZ/nZ
¯
kUn
k n
¯
k(Z/nZ,+)
φ(n)Unφ(n)n
φ(n) = card{k: 1 kn, k n= 1}
p s φ(ps) = psps1
m n
f:Z/mnZZ/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)¯
kUnmZ¯
k¯m=¯
1n|km 1
sZkm 1 = sn km sn = 1 k m
(ii)(iii)kn= 1 m, s Zkm +sn = 1 m¯
k=¯
1
tZ¯
t=tm¯
k¯
k(Z/nZ,+)
(iii)(i)¯
k(Z/nZ,+) mZm¯
k= 1 ¯m¯
k=¯
1
¯
k
φ(ps)k1k < ps
psE={1,2, . . . , ps}F={kE:p|k}φ(ps) = card(E)card(F) =
pscard(F)F={p, 2p, 3p, . . . , ps=ps1p}(F) = ps1φ(ps) = psps1
m n
f:Z/mnZZ/mZ×Z/nZ
¯x[mn]7→ (¯x[m],¯x[n])
f¯x[mn] = ¯y[mn]mn |xy m n
m|xy n |xy¯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|xy n |xy m n
mn |xy¯x[mn] = ¯y[mn]f
¯xUmn 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
ipki1
i)
Gn={zC|zn= 1}
Gn(C,·)
G n (C,·)G=Gn
G
Gn(C,×) 1 Gnu, v Gn(uv1)n=
un(vn)1= 1 uv1Gn
Gnz=exp(2i
n) = exp(2πi
n)kGn=grhξi={1, ξ, ξ2, . . . , ξn1}
ξ=exp(2πi
n)
G n (C,×)zG zn= 1
GGn|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 CuKerf 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= 2i k Zexp(z) = 1
Kerf = 2πiZ
z=a+bi Imf z 6= 0 z=ρeu=c+di exp(u) = ecedi =ρe
c= ln(ρ)d=θ2π z uCf
f=C
G x 7→ x1G G
x, y G(xy)1=y1x1= (yx)1xy =yx G
2(R)A=0 1
11B=01
1 0 A, B
AB
A2=11
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, kNAB
G g G γg:GG γg(x) =
gxg1,xG
γgG g
γgh =γgγh(γg)1=γg1
(G)G(G)
G
(G) (G)
(G)
=G/Z(G)Z(G)G
x, y G γg(xy) = gxyg1=gxg1gyg1=γg(x)γg(y)γg
γgγg1=γg1γg=IGγg
xG γgh(x) = ghx(gh)1=ghxh1g1=γg(γh(x)) = γgγh(x)γgh =γgγh
γgγg1=γe=IGγg1= (γg)1
IG(G)γg, γ1
h(G)γgγ1
h=γgh1(G) (G)
(G)
σ(G)xG σ γgσ1(x) = σ(1(x)g1) = σ(g)(g1) =
γσ(g)(x)σγgσ1== γσ(g)(G) (G)
(G)
φ:G(G)φ(g) = γgφ(gh) = γgh =γgγh=φ(g)φ(h)φ
G/ φ
=(G)gG
gφγg=IG⇔ ∀xG, gxg1=x⇔ ∀xG, gx =gx gZ(G)
f:GG0xG 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 =xyxx1=γx(yx)γxx
o(xy) = o(γx(yx)) = o(yx)
xyz =xyzxx1=z1zxyz
m n
Z/n.Z×Z/m.Z
=Z/nm.Z
φ:ZZ/n.Z×Z/m.Zφ(x) = (π1(x), π2(x)) π1(x)π2(x)
x n m φ
xZx(φ)n|x m|xmn|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
gG gHg1G H |gHg1|=|H|
G H m H G
gG γg:GG γg(x) = gxg1
gHg1=γg(H)H
γg|gHg1|=|γg(H)|=|H|
G H m g G|gHg1|=|H|=m
gHg1=H H CG
G e xG 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
gG g2H
a1Ha H a G a H a /H
[G:H] = 2 H(G/H)g={H, Ha}
H(G/H)d={H, aH}G=HHa =HaH Ha =G\H=aH
a1Ha H
gG g H g2H g /H g2/H g2Hg G =HHg
g2=hg h H g =hH g2H
G={e, a, b, c}e
a2=b2=c2=e ab =ba =c ac =ca =b bc =cb =a G
G
5
xG 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
G5
• |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 :xG} B(E)E
aG ρa:EE ρa(xH) = axH
ρaρa∈ B(E)
GΦ : G→ B(E)a7→ ρa
N
G/N B(E)
N G H
H m
[H:N] (m1)!
m|G|H
G
aG xH =yH x1yH(ax)1(ay) = x1a1ay =x1yH
axH =ayH ρa
G×EE(g, xH)7→ gxH G E
xH yH ρa(xH) = ρa(yH)axH =ayH (ax)1(ay)H(ax)1(ay) = x1y
xH =yH ρa
yH E ρa(a1yH) = yH ρa
ρaE
a, b, x G ρab(xH) = abxH =ρa(ρb(xH)) ρab =ρaρbΦ(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 x1gx K K H x1gxH =H gxH =xH
gN
G/N B(E)E= [G:H]B(E)
=Sm
|G/N|= [G:N] = [G:H][H:N]|m! [H:N]|(m1)!
[H:N] = 1 p[H:N]p|(m1)!
p k = 1, . . . , m 1pm1< m p
|G|pm 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|/|HN| |H| |N|
HN={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 =a1
NCG bab =bab1=akN a =bbabb =bakb= (bab)k= (ak)k=ak2
ak21=e q |k21 = (k1)(k+ 1) q q |k1q|k+ 1
k∈ {0,1,2. . . q 1}k= 1, k =p1k= 1 ab =ba G
k=p1bab =ap1=a1G
(G, ·)e G 6={e} {e}G
G
G
G
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