ΥڡϤƤʥ֥åޡɲΥڡޤϤƤʥ֥åޡ Υڡlivedoor åפɲΥڡޤlivedoor å

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[]G1鷲G2ؤμfƱǤȤϡG1Ǥդθx,yФơΩĤȤǤ롣
f(xy)=f(x)f(y)

[]GNGʬȤ롣GG/Nؤμff(x)=xNˤäơСf:GG/NפϽƱǤ뤳Ȥ򼨤

[]fƱǤ뤳Ȥ򼨤ˤϡf(xy)=f(x)f(y)ΩĤȤǧФ褤

()
=f(xy)
=xyN

()
=f(x)f(y)
=(xN)(yN)
=xyN(G/NѤ)

äơ()=()ꡢfϽƱǤ롣

[]Ʊñ̸ñ̸˰ܤոո˰ܤ

[]G1,G2򶦤˷Ȥ롣

[1]Ⱦ򼨤

G1ñ̸e1G2ñ̸e2Ȥ롣

f:G1G2פƱǤȤȡf(e1)=e2ΩĤȤ򼨤Ф褤

G1ˤơe1e1=e1Ωġ

fǰܤȡf(e1)f(e1)=f(e1)Ȥʤ롣

f(e1)G2θǤ뤿ᡢG2ˤf(e1)εո򱦤餫ȡf(e1)=e2Ȥʤ롣

[2]Ⱦ򼨤

aG1ǤդθȤȤf(a-1)=(f(a))-1򼨤Ф褤

G1ˤơaa-1=e1Ωġ

fǰܤȡf(a)f(a-1)=f(e1)Ȥʤ롣

Ⱦη̤ꡢf(a)f(a-1)=e2

G2ˤơG2θf(a)εո򺸤餫ȡf(a-1)=(f(a))-1Ȥʤ롣

ˡ󷲤ؤμ

[]G򷲡aG1ĤθȤ롣
ˡZ鷲Gؤμff(x)=axСf:ZGϽƱˤʤ롣

[]ZǤդθx,yФơΩġ

f(x+y)
=ax+y
=(ax)(ay)
=f(x)f(y)

äơfϽƱǤ롣

Ʊγˤ

[]f:G1G2G1鷲G2ؤƱȤ롣
G1θfˤäơG2ñ̸˰ܤΤνȤKer(f)ɽ
ޤG2θf(x) (xG1)ηɽΤνfȤf(G1Im(f)ɽ

ȤäɽȼΤ褦˽񤱤롣

Ker(f)={xG1|f(x)=e2}

ˤ

[]f:G1G2G1鷲G2ؤνƱȤ롣
ΤȤfγKer(f)ϡG1ʬǤ롣

[]

[1]fγKer(f)G1ʬǤ뤳ȤΤ롣

  1. x,yKer(f)xyKer(f)Ωġ
    • x,yKer(f)Ȥȡx,yG1, f(x)=e2,f(y)=e2Ǥ롣
    • xyG1, f(xy)=f(x)f(y)=e2e2=e2ǤΤǡxyKer(f)ˤʤ롣
  2. e1Ker(f)Ωġ
    • f(e1)=e2Ǥ뤫顣
  3. xKer(f)x-1Ker(f)
    • xG1, f(x)=e2, f(x-1)=(f(x))-1=e2-1=e2Ǥ뤫顣

äơKer(f)G1ʬˤʤ롣

[2]Ker(f)G1ʬǤ뤳ȤΤ롣

ʬꡢgG1,nKer(f) g-1ngKer(f)פΩĤȤǧФ褤

f(g-1ng)
=f(g-1)f(ng)ʢfϽƱǤ뤫
=f(g-1)f(n)f(g)ʢfϽƱǤ뤫
=f(g-1)e2f(g)ʢnKer(f)ꡢf(n)=e2
=f(g-1)f(g)ʢe2G2ñ̸Ǥ뤫
=f(g-1g)ʢfϽƱǤ뤫
=f(e1)ʢe1G1ñ̸Ǥ뤫
=e2ʢ[]ֽƱñ̸ñ̸˰ܤոո˰ܤפȾ

äơg-1ngKer(f)

ʾꡢKer(f)G1ʬǤ롣

[]f:G1G2G1鷲G2ؤνƱȤ롣
ΤȤff(G1)ϡG2ʬǤ롣

[]f(G1)G2θʤ뽸Ǥ뤿ᡢf(G1)G2ʬǤ뤿ˤϡ3򼨤Ф褤e2G2ñ̸Ȥ롣

  1. x,yf(G1)xyf(G1)
  2. e2f(G1)
  3. xf(G1)x-1f(G1)

[1]x,yf(G1)xyf(G1)򼨤

xy
=f(a)f(b)ʢx,yf(G1)Ȥȡx=f(a),y=f(b), a,bG1Ȥ˽񤱤뤫
=f(ab)ʢfϽƱǤ뤿
f(G1)

[2]e2f(G1)򼨤

e2
=f(e1)ʢ褳G1ñ̸e1Ȥơ[]ֽƱñ̸ñ̸˰ܤոո˰ܤפȾ
f(G1)

[3]xf(G1)x-1f(G1)򼨤

x-1
=(f(a))-1ʢx=f(a), aG1֤
=f(a-1)ʢ[]ֽƱñ̸ñ̸˰ܤոո˰ܤפθȾ
f(G1)

ʸ

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