代数学引论 第二版 (聂灵沼 丁石孙 著) 高等教育出版社 课后答案 8页

'课后答案网：www.hackshp.cn若侵犯了您的版权利益，敬请来信告知！www.hackshp.cn课后答案网您最真诚的朋友www.hackshp.cn网团队竭诚为学生服务，免费提供各门课后答案，不用积分，甚至不用注册，旨在为广大学生提供自主学习的平台！课后答案网：www.hackshp.cn视频教程网：www.efanjy.comPPT课件网：www.ppthouse.com课后答案网www.hackshp.cn 课后答案网：www.hackshp.cn若侵犯了您的版权利益，敬请来信告知！www.hackshp.cnhklgdemjifb~ZB
%,C!S1.G09&{N⊳G,|N|=|G/N|C
=>a#F|N|,Aa∈N.`XaS|π:G−→G/n,a7−→a.4+o(a)|o(a),:o(a)||N|.o(a)||G/N|,49o(a)=1,a∈N.khdaw.com2.c{G05rs#>r(r,s)=1,HjcY1c=ab,ra#r,b#s,va,b(c#0h
`8
0#0)Fx,y∈Z,"xr+ys=1,4+c=xrysys,b=cxr.(c)(c).`a=c8a#=GFkCA0x=alt90S
k.`H=,f:H−→H,x7−→x4(k,|H|)=1,:m,kkmkmk"mk≡1mod|H|.4+x=y=⇒x=y=⇒x=y,f
k=a;H9&{:课后答案网f;dxHl9vT90S
4.Hj0{>ab=ba9"#
−1(ab)a,`=o(ab)<∞,A8ba=aY"o(ab)=o(ba);=o(ab)=∞,nn−1n)<}AGFn,(9(ab)6=1,+(ba)=(aaba)=a−1(ab)na6=1,o(www.hackshp.cnba)=∞.5.n>2.Hj9&{Gn#>#5n
−1,−1`n>2o(a)=n,Aa6=avo(a)=o(a)=n.4Gn#>)%:5n
6.n>2HjZ(Sn)={e}.`8C&^9u#3π∈Z(Sn),|9π=1,πf5>Æ"
7.Hj9^-{Q+#}09&#R{-E{
`G=Qα1⊕Qα2⊕···⊕Qαn,α1,α2,···,αn∈Q.αi=bi/ai,(ai,bi)=1,ai,bi∈Z.a1,a2,···,an#U(7m,2
G≤Z·1.m1khdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com 课后答案网：www.hackshp.cn若侵犯了您的版权利益，敬请来信告知！www.hackshp.cn8.G059&#F{
=G#f5>9&>AG9&{
`G=≤···,A|G|≤||||···||=o(a1)o(a2)···o(an)<∞.10.HjSn8n−15)F(12),(13),···,(1n)Sn/Y8n−15)F(12),(23),···,(n−1n)
`)<∀σ∈Sn,σY13SÆ"#bF#H
4,XabF#>Y
(1)(i1i2···im)05bFA(i1i2···im)=((i1i2)(i1i3)···(i1im)),+(i1i2)=(1i1)(1i2)(1i1),4
f05)F(ij)=(1i)(1j)(1i).:f05bFY8(12),(13),···,(1n)+SnY8g
(2)j=i+1,A82,HjAn8(123)=(12···n)(ns#w+),An8(123)=(23···n)(nn#w+).`℄11Y
(234),(345),···,(n−2n−1n)Y
(k−1kk+1)(4≤k,k+1≤n)+(12k)=(12k−1)Y(1jk)=(1j−1k)(j−1jj+1)(j+1≤k)(i−1ii+1)(i+1≤j)Y(ijk)=(i−1jk)Y
4+AnY
13.σ=(12···n),HjσSnl#)DR<σ>,HjσSn#8X℄>课后答案网(n−1)!5>
`C(σ)σSnl#)DR$|<σ>≤C(σ).yτ∈C(σ),"Hτ∈<σ>.τ4(12···n)=(τ(1)τ(2)···τ(n))=(12···n),τ(i)=i,A9(τ(1)τ(2)···www.hackshp.cnτ(n))=(ii+1···n12···i−1),τ(1)=i,τ(2)=i+1,···,τ(n−i+1)=n,τ(n−i)=1,···,τ(n)=i−1,/!123···n−i+1n−i···nτ=∈<σ>.ii+1n···12···i−116.H1,H2{G#_5R{HjH1∩H2#}0VoH1#05Vo=H2#05Vo#
`)1.HjG>90Fn!#2p.G;R{GDG9|G:H|=p,:|G:N|=p,N=H⊳G.21.Hj}02F#6{9G3R{
819
H⊳G.yG#课后答案网2R{K=,Y
HK=G.8=∼=S322.&zCÆ9#15{
`yP∈Sylwww.hackshp.cn3(G),Q∈Syl5(G),$|Q⊳G.n3G#Sylow-3R{58&^18&^17#3Y
n3=3m+1vn3||G|,qn3=1,P⊳G.9G=P×Q.P=,Q=,ATrxy∈Gvo(xy)=15,Y
G=-E15{9℄905
23.&zCÆ9#10{
`|G|=10,yP∈Syl2(G),Q∈Syl5(G),Y
Q⊳G.n2G#Sylow-2R{58,P=,52b−1,b−1AG==∼=D10ra=a4Q⊳G=⇒ab∈Q.bbi;4G2Fvo(a)=o(a),+a=a,i=2,3,4.3khdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com 课后答案网：www.hackshp.cn若侵犯了您的版权利益，敬请来信告知！www.hackshp.cn2−1−1−1ii28a=b(bab)b=b(a)b=a,4+i≡1mod5,9i=4.|G|=10GT9_59℄*10-E{10,i{
25.2p,qÆ#Hjpq{>05G;#SylowR{
`yP∈Sylp(G),Q∈Sylq(G).(1)p>q|G:P|=q,Y"P⊳G.(2)p09mq+1|p=⇒mq|(p−1)(p+1),+q=p+1,9q=3,p=2.G945Sylow-3R{4+G9853>352>05>
:G#Sylow-2R{0G>05G;#SylowR{
26.V_59F{G,3ÆB(i)a3=e,)9#a∈G;(ii)a4=e,khdaw.com)9#a∈G.33[Y(i)G=.(ii)G=D8GDG=Q8.27.H,K|K:H∩K|5H#:o
28.=9&{G9052p.#-E#Sylow2-R{AG9052#R{
课后答案网`XaG#:GAW7Y
G#W7#vG−→Sn(n=|G|)
Æ1G≤Sn.P=<σ>∈Syl2(G),Y"o(σ)nσSnsF
4+GAn=Sn,+G/G∩An≃GAn/An2-E{:|G/G∩An|=2.www.hackshp.cn29.G9&{A,BG#_52[RB=|A|+|B|>|G|,AAB=G.−1−1`=AB6=G,A9g∈Gg/∈AB.`B={b|b∈B},−1−1AgB∩A=∅.4A9a∈Ava∈gB,Ab∈B,"a=gb−1=⇒g=ab∈AB,−1e*
4+|A|≤|G|−|gB|=|G|−|B|,Y"|A|+|B|≤|G|,e*
30.H9&{G#052p.#R{HjG6=∪gHg−1,g∈G.Sgg`(1)HEG)<∀g∈G,H=H,:G6=H=g∈GH.4khdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com 课后答案网：www.hackshp.cn若侵犯了您的版权利益，敬请来信告知！www.hackshp.cn(2)H5GNG(H)GvH#8XR{5t?|G:NG(H)|,Sg|g∈GH|≤|G:NG(H)|(|H|−1)+1=|G:NG(H)||H|−|G:NG(H)|+1≤|G|−|G:NG(H)|+1<|G|.31.W&S4#Sylow2-R{Sylow3-R{
|=24=3·23.[Y|S4(1)2
Sylow-3R{9453<(123)>,<(124)>,<(134)>,<(234)>.(2)2HK={(1),(12)(34),(13)(24),(14)(23)}⊳S4.+<(12)>K,<(13)>K,<(14)>Kt8R{
<(12)>K={(1),(12)(34),(13)(24),(14)(23),(12),(34),(1423),(1324)}.<(13)>K={(1),(12)(34),(13)(24),(14)(23),(13),(24),(1432),(1234)}.<(14)>K={(1),(12)(34),(13)(24),(14)(23),(14),(23),(1342),(1243)}.32.HjÆ56G148#{
khdaw.com`4148=4·37,148{#37R{G;:ÆYm{
)<56=7·8,=Sylow-7R{0AG2{=Sylow-7R{Æ0A8985Sylow-7R{
EjG94857>Sylow-2R{0:ÆYm{
33.G09&{N⊳G,PN#05SylowR{TPG#G;DR
HjG=NT.gg`)<∀g∈G,A9N=N=⇒P∈N.4+n∈N,"n−1Pg=Pn=⇒Pg=P.−1gn∈T=⇒g∈nT⊂NT,+G=NT.34.G09&{PG#05Sylowp-R{N⊳G.HjP∩NN#05Sylowp-R{PN/NG/N#05Sylowp-R{
`|N:P∩课后答案网N|=|NP:P|||G:P|,:P∩N∈Sylp(N).|G/N:PN/N|=|G:PN|||G:P|,:PN/N∈Sylp(G/N).35.G09&{H−1∩Ha∈GaPaH#05Sylowp-R{
`P∩Hwww.hackshp.cnH#05p-R{:R∈Sylp(H),"P∩H≤R.aa;RG#p-R{4+a∈G,"R≤P.2"R=P∩H.3,p37.{G#p0
HjG2FAG(1)=Z(G).2`4Z(G)>1,1G/Z(G)p{GDp{
+G/Z(G)′′FY"G≤Z(G).8G2FY"1G≤Z(G)G.8392,
G/Z(G)ÆYmp{:9|G/Z(G)|=p|Z(G)|=p.4+|G′|=p,′Y"G=Z(G).n,N⊳G,|N|=p.38.G0p-{|G|=pHjN=.4N≤Z(G),:G#>YF+GF
40.Hj}09&{(92#9&{9_5S9
−1`GF6f:G−→G,g7−→gS9
;8<|G|>2,:Y3_w6khdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com 课后答案网：www.hackshp.cn若侵犯了您的版权利益，敬请来信告知！www.hackshp.cn(1)G9>(2>
f=1,G$F2-{
|G|=2n,nnnn−1AGLn(2)≃Aut(G),|Aut(G)|=(2−1)(2−2)···(2−2)>2.(2)G905>Æ2>
f6=1,{f,1}_5S9
G2FG/Z(G)≃Inn(G),G>Z(G),:|Inn(G)|≥2,+|Aut(G)|≥2.48.Hj>{V={e,(12)(34),(13)(24),(14)(23)}#S9{=S39
`847
|Aut(V)|=6.σ:(12)(34)7−→(13)(24),(13)(24)7−→(12)(34),(14)(23)7−→(14)(23).τ:(12)(34)7−→(14)(23),(14)(23)7−→(12)(34),(13)(24)7−→(13)(24).<σ>,<τ>(Aut(V)#,R{v<σ>6=<τ>,:Aut(V)2khdaw.com-E+qAut(V)≃S3.52.Hj{An,N>4f53-bF(Y10FR
(1)8HjAn=An.`(ijk)=(mij)(jnk)(mji)(jkn)=[(mji),(jkn)],4+′,′An≤An+An=An,n>4.55.05{G8>a,b+a,bÆBa4=b4=1,a2=b2,b−1ab=a−1.HjG08{v2F#zR{(G;
`A=,B=,AG==AB,+|AB|=|A||B|/|A∩B|=8.课后答案网$|G2F
8<4R{2,:rG;
GT9052>:2R{G;
www.hackshp.cn7khdaw.com若侵犯了您的版权利益，敬请来信通知我们！℡www.khdaw.com'