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[]G1,G2򶦤˷Ȥf:G1G2ƱȤ롣
ΤȤfγKer(f)ϡG1ʬǡG1/Ker(f)ff(G1)ƱǤ롣

G_1~/~\mathbb{Ker}~f~\conq~f(G_1)

ñΤˡKer(f)=KȤȡΩġ

G_1~/~K~\conq~f(G_1)

ǡ\bar{f}:G_1~/~K~\rightarrow~f(G_1)\bar{f}(xK)~=~f(x)~\,~(\in~f(G_1))
Ʊ򼨤ˤϡΡfƱǤ뤳ȤΤФ褤

;xKɽxμˤʤ

xʳɽyäȤ롣¨xK=yKȤ롣
ȡyxKθǤ뤿ᡢy=xk, kKȽ񤱤롣 ΤᡢΩġ

f(y)
=f(xk)ʢy=xk
=f(x)f(k)ʢfϽƱ
=f(x)e2ʢKfγˤf(k)=e2
=f(x)

ǡxK=yKע͡f(x)=f(y)פ롣

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[1]fϽƱǤ뤳Ȥ򼨤

  • f( (xK)(x'K))=f(xx'K)=f(xx')
  • f(xK)f(x'K)=f(x)f(x')=f(xx')

äơf( (xK)(x'K))=f(xK)f(x'K)ΩĤᡢfϽƱǤ롣

[2]G1/Kΰۤʤ븵fˤäƱ˰ܤ뤳ȤϤʤȤ򼨤

֡f(xK)=f(x'K)ע͡xK=x'Kפ򼨤Ф褤

f(xK)=f(x'K)
f(x)=f(x')
(f(x))-1f(x')=e2
f(x-1)f(x')=e2ʢfƱǤСոϵո˰ܤ
f(x-1x')=e2ʢfϽƱǤ뤫
f(k)=e2ʢk=x-1x'Ȥ
kKer(f)=K

k=x-1x'
xk=xx1x'ʢ躸xݤ
xk=e1x'ʢxx1=e1
xk=x'

äơx'K=xKΩġ

[3]f(G1)θϤ٤ơf(x) (xGSUB{1}/K;)ηɽ뤳Ȥ򼨤

f(G1)θf(x) (xG1)η˽񤱤Τǡf(x)=f(xK)Ȥʤ뤫顢f(G1)θ٤ơf()ηɽ롣

äơ[1][2][3]ꡢfƱǤ뤳ȤΩġ

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