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ܼ

Ϥ

GHGʬȤ롣ΤȤgH{gh;hH}ν礽ΤΤǤꡢgޤHκ;ࡢHg={hg;hH}gޤHα;Ǥ롣;νG/H;νH\Gɽ

;ƱΤα黻(aH)(bH)=(ab)Hơꤿ

ΤȤ黻ɽμˤʤȤΤʤФʤʤ
ŪˤȡΩĤȤǤ롣

aH=a'H,bH=b'HעΡ(ab)H=(a'b')H

줬ΩĤˤϡǤդa,bG;h,h'HФơ(ah)(bh')=(ab)hhH¸ߤ뤳ȡפɬ׽ʬǤ롣

δطơʬ롣

ʬ

[]HϷGʬǤȤ롣
\forall~a~\in~G,~\forall~h~\in~H;~a^{-1}ha~\in~HȤˡʬHGʬnormal subgroupȸƤ֡

[]ʬξϼΤ褦Ѥ뤳ȤǤ롣ƱͤǤ롣

  1. aG;Ha=aH
  2. a,bG;alb (mod H)arb (mod H)
  3. a,b,c,dG;alb (mod H)cld (mod H)aclbd (mod H)
  4. aG;aHa-1H
  5. aG;aHa-1=H

Ha=aHȤha=ah(hH)*1̣ΤǤϤʤha=ah'(h,h'H)ȤʤHθh,h'¸ߤ뤳Ȥ̣Ƥ*2

[]

[1](1)(2)

ƱƱΩġ

[2](2)(3)

alb (mod H)
arb (mod H)
acrbc (mod H)
aclbc (mod H)(*)

ޤƱͤˤcld (mod H)bclbd (mod H)(**)

(*)(**)ȿΧꡢaclbd (mod H)Ωġ

[3](3)(4)

baHaH=bHalb (mod H)ꡢ(3)̿ϼΤ褦˽񤭴롣baH,dcHabacHǤ롣Ĥޤꡢa,cG;aHcHacHǤ롣

c=a-1ȡ

aHcHacH
aHa-1Haa-1H=eH=H
aHa-1H

ޤaHa-1aHa-1HʤΤǡaHa-1H

[4](4)(5)

(4)aǤդʤΤǡa-1ǹͤȡa-1HaHΩġ

HǤդθhȤȡa-1haHǤ뤫顢HΤ븵h'äơa-1ha=h'Ȥʤ롣äơh=ah'a-1aHa-1Ȥʤ뤫顢HaHa-1Ǥ롣

ޤ(4)Ȥ߹碌ơaHa-1=HΩġ

[5](5)(1)

aHǤդθah(hH)ɽΤǡ

aha-1
aHa-1
=Hʢ貾(5)

äơaha-1=h'HΩġ

ah=h'aHaȤʤΤǡaHHaΩġƱͤˤHaaH⼨롣

äơaH=HaΩġ

[]GĴʤ顢٤ƤʬʬǤ롣

[]GĴΤȤϡaG;aH=HaΩġ

椨ˡĴʬϤ٤ʬǤ롣

[](Z,+),(Zn,+),(Q*,),(RSUP{*},),(C*,),(Mn(K),+)ʤɤϤ٤ƲĴǤΤǡʬϤ٤ʬǤ롣

㤨СZʬmZʬǤ롣

[]CnüSLn(C)ϡnGLn(C)ʬǤ롣

ʬפǤͳϡʬHξ;ƱΤ黻뤳ȤǤȤǤ롣

[]S3ʬǤϡA={,,2ϡ;ȱ;ϰפ롣ĤޤꡢΩġ

  • S=A+A
  • S=A+A

äơAS3ʬǤ롣

[]

  • G/Hñ̸eH(=H)Ǥ롣
  • G/Hεո(aH)-1=(a-1)HǤ롣

ʬ

[]GˤơG濴Z(G)ʬǤ롣

[]GʬHλؿ2ʬʬǤ롣
¨[G:H]=2Hʬ

[]G=H+gH=H+HgϡGα;Ⱥ;ؤʬǤ롣

ʬ϶̸ʤΤǡgH=HgǤʤФʤʤäơHʬǤ롣

Ϥ褯Ȥ롣

[]νƱաGHγKer()GʬǤ롣

[]Ǥդ˦ǢKer(),aGȤȤ

(aa-1)
=(a)()(a-1)ʢդϽƱǤ뤫
=(a)(a-1)ʢKer
=1

äơaa-1Ker()Ωġ

ʬ

[]ʬˤʬͭʬȸäƤ롣ȯ뤳Ȥˤꡢ

[]GμʬϤɤGʬǤ롣

[]GμʷȤGȤȡñ̷{e}ΤȤǤ롣

[1]N=GΤȤ

g-1ngG=NǤ뤿ᡢGGʬˤʤ롣

[2]N={e}ΤȤ

g-1ng
=g-1egʢNθeʤᡢn=e
=g-1g
=e
N

äơNGʬˤʤ롣

[]оηS3ˤơʬN=<(1 2 3)>={(1), (1 2 3), (1 3 2)}ʬˤʤ롣

N
=<(1 2 3)>
=<d>ʢ跲ɽ
={d,d2,d3}
={e,d,f}ʢ跲ɽ顢d2=fd3=e

ϡn{e,d,f}(=N)g{a,b,c,e,d,f}(=G=S3)줾ǡg-1ngNǤ뤳Ȥǧ롣

  • n
    • n=eΤȤ
      • g-1eg=g-1g=eN
    • n=dΤȤ
    • n=fΤȤ
  • g
    • g=aΤȤ
    • g=bΤȤ
    • g=cΤȤ
    • g=eΤȤ
      • n,gɤNθǤꡢ黻ˤĤĤƤ뤿ᡢg-1ngN餫
    • g=dΤȤ
      • n,gɤNθǤꡢ黻ˤĤĤƤ뤿ᡢg-1ngN餫
    • g=fΤȤ
      • n,gɤNθǤꡢ黻ˤĤĤƤ뤿ᡢg-1ngN餫

ĤΡn{d,f}g{a,b,c}Ȥ߹碌6ѥǡg-1ngNǤ뤳Ȥǧ롣

  • n=dg=aΤȤ
    • a-1da=aba=ba=fN
  • n=fg=aΤȤ
    • a-1fa=afa=ca=dN
  • n=dg=bΤȤ
    • b-1db=bdb=cb=fN
  • n=fg=bΤȤ
    • b-1fb=bfb=ab=dN
  • n=dg=cΤȤ
    • c-1dc=cdc=ac=fN
  • n=fg=cΤȤ
    • c-1fc=cfc=bc=dN

ʾꡢɤξǤg-1ngNΩĤᡢNS3ʬǤ롣

<(1 2)>={(1), (1 2)}S3ʬˤʤ뤬ʬˤϤʤʤ

ʤʤС<(1 2)>={e,b}ǤꡢΤ褦˷׻Ǥ롣

a-1ba
=abaʢ跲ɽ顢a-1=a
=da
=c

c<(1 2)>°ʤǤ롣

GʬHǾ;ʬ䤹Ȥ;ȤΤȱ;ȤΤȤǤϷ̤ۤʤ뤳Ȥ롣HʬΤȤϡ;ȱ;बפ롣ĤޤꡢɤΥ饹ʬȤäƤ⡢Ʊ饹ʬˤʤ롣

[]NGʬʤСGǤդθaФơaN=NaΩġ

嵭μκդ;ࡢդ;Ǥ롣

[]

[1]aNNa򼨤

aNθan (nN)ȤɽȤǤΤǡΩġ

an
=ana-1a(e=a-1a) =(a-1)-1na-1a
Na(NGʬǡa-1GnNǤꡢ(a-1)-1nNǤ뤫)

äơaNθϤ٤Na°롣

[2]NaaN򼨤

Naθna (nN)ȤɽȤǤΤǡΩġ

na
=aaSUP{-1}na(e=aa-1)
aN(NGʬǤ뤿ᡢa-1naNΩĤ)

äơNaθϤ٤aN°롣

äơ[1][2]ꡢaN=NaΩġ

ʬǤϺ;ȱ;बפ뤿ᡢñ;Ȥ

ʬľ

[]G1,G2H1,H2줾ʬȤ롣
H_1~\times~H_2G_1~\times~G_2ʬǤꡢΩġ
(G_1~\times~G_2)~/~(H_1~\times~H_2)~\simeq~(G_1/H_1)\times(G_2/H_2)

[]ͽƱȤ߹碌ơͽƱG_1~\times~G_2~\to~(G_1/H_1)\times(G_2/H_2)СγˤH_1~\times~H_2Ǥ롣

ñ㷲

[]ñ㷲
Gʬ{e}GʳʬʤȤGñ㷲Ȥ

ʬʳΥå

ʬʳΥåȤϤ켫ΤǤϷǤϤʤ

[]Gʬä硢GHʬ򤷤åȤϷˤʤ롣

[]HʬǤȤϡGΤ٤ƤθФơH=HΩĤȤ̣롣

GʬG=H+H+H+H+ĤȤȡΩġ

HH
=HHʢH=H
=H

Ȧ¤GθʤΤǡ¤GθǤ롣ĤޤꡢHϥåȤΤɤ줫ˤʤ롣

;ñ̸HǤ롣ޤHεոϦ-1HȤʤ롣

äơ

H-1H
=-1HH
=H
=H

ʬˤ;

NGʬȤȤGNˤ;11Ĥ򸵤ȹͤơʽ򸵤ªˡ;Τκ뽸G/NȤɽ

[]GNˤäơ3ĤΥ饹ʾ;N,aN,bNʬ줿Ȥ롣

ΤȤ

G/N={N,aN,bN}

Ωġ

ĤޤꡢG/NϽνǤ롣

G/N˱黻

G/N˼ˡѤ롣

(aN)(bN)=abN

aN=a'NbN=b'Nע͡abN=a'b'NפΤƤɬפ롣

[]NGʬΤȤG/N(aN)(bN)=abNѤСG/NϷˤʤ롣

G/NϷˤʤ뤬ηGNˤȤ

ʸ

  • ؾʳؤΤ4ϡ3ηˡpp.57-67
  • طġ2ϡ5ʬ;pp.109-112
  • ؤ֤ä73ϡγǰ3.5ʬ;ෲpp.56-57
  • ضΩֺ¡21ο8ĤΤ
  • طʤ󤫤勞ʤ
  • ر


*1 줬ΩĤȤȤ
*2 ΤȤ顢餫˶ʤʬξȤ롣